Lectures and exercises Москва Ижевск 2005 УДК 531.1 ББК 22.21 П50 Polishchuk D.F., Krylov E.G. Integrational mechanics. <...> In the book the basic ideas of integrationalmechanics 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. <...> SYSTEM APPROACH IN STATICS AND KINEMATICS 22 2.1. <...> Information operator of null action and axioms of statics . . 22 2.2. <...> 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 acceleration in a rigid body plane motion . . . . . . 34 2.16. <...> All axioms and theorems of statics are essentially compressed by means of the application of informational operator of null action. <...> 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. <...> This compact incorporates also special compacts on vibration and damping,stability and impact. <...> Special classification of “physical bodies” is offered on the base of the general operator of infirmation. <...> 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. <...> Lagrange’s desire was to intensify the mathematical basis in mechanics and formalize the deriving of initial <...>
Integrational_Mechanics._Lectures_and_exercises..pdf
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»,
2005. — 148 с.
In the book the basic ideas of integrationalmechanics 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 5-93972-372-1
Polishchuk D.F., KrylovE.G., 2005
c
http://rcd.ru
http://ics.org.ru
ББК 22.21
Стр.2
Contents
Introduction .. .. ... .. .. ... .. ... .. .. ... .. ... 5
Chapter 1. CLASSICAL MECHANICS AS A SYSTEM THEORY .7
1.1. Structure of the course of classical mechanics ... ... .... . 7
1.2. Unity of mathematics, physics, and philosophy in Newton’s mechanics
.... .... ... .... .... .... ... .... . 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. Information operator of null action and axioms of statics .... . 22
2.2. System of concurrent forces .... .... .... ... .... . 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. The basic theorem of statics .... .... .... ... .... . 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. Invariants of force system . .... .... .... ... .... . 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 compression
principle . .... ... .... .... .... ... .... . 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 acceleration
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
Стр.3
4
Contents
Chapter 3. DYNAMICS .. .. ... .. ... .. .. ... .. ... 44
3.1. Newton’s laws systematization ... .... .... ... .... . 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. Energy mechanics ... ... .... .... .... ... .... . 72
3.6. Elements of Lagrange’s analytical mechanics ... ... .... . 79
3.7. Compact “Impact phenomena in mechanics” ... ... .... . 82
3.8. Compact “Linear and nonlinear problems in dynamics” .... . 89
3.8.1. Classifications of vibration problems .... ... .... . 89
3.9. Compact “Stability” . . . . . .... .... .... ... .... . 100
3.10. Analytical mechanics as an “ideal” theory . .... ... .... . 107
3.11. System classification of forces ... .... .... ... .... . 112
3.12. Classification of “physical” bodies . .... .... ... .... . 115
Chapter 4. EXAMPLES OF PROBLEM ANALYSIS . ... .. ... 119
4.1. Kinematics of mass – point particle. Analogies .. ... .... . 119
4.2. Dynamics of mass – point particle .... .... ... .... . 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. Bearing reactions ... ... .... .... .... ... .... . 139
4.8. Differential equation of motion of a mechanism . . ... .... . 143
Bibliography . . . ... .. .. ... .. ... .. .. ... .. ... 146
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