Abstract
to the web
About
author Konoplev V.A.
Preface
to
the web
About
NavieStoks equations
FULL
TEXT BOOKS:
Algebraic
methods in Galilean mechanics Konoplev
V.A. (PDF
eng)
(PDF
rus)
Aggregative
mechanics of rigid body_systems Konoplev
V.A.
(PDF
eng) (PDF
rus)
Computer
technologies in research of multichain mechatron systems
(Konoplev
V.A.
together with Gavrilov S.V.)
(PDF
rus)
Additional
chapters:
Questions
of geometrical optics and dinamics of radiotelescope with
hyperbolical and elliptical contrreflectors Konoplev
V.A.
(
PDF rus)
MATLAB – programs of computer modeling of
geometrical optics questions
rtel.m
rtg.m

Preface
This
site attempts to build fragments of theoretical
mechanics by
replacing the standard working
material (a finite set of massive points without volume) by
an infinite continuous set of points without mass and
without volume. The main goal is to take the first steps in
creating a discipline called digital mechanics;
In the
classical case, before a concrete calculations, it is necessary to
perform many tedious analytical
transformationsand preparatory work. For example, the
translation of computational dependencies from the language
of tensor analysis and the algebra of geometric
vectors to
the language of scalar equalities, etc. Developed
in AMMG (the
first book of the site), the mathematical models of
mechanics (including a convenient system of physically
understandablenotation) allows you to perform all analytical work
( the formulation of concepts, the proof of theorems,
the creation of Matlabprograms, etc.) on a computer using
standard text editors.;
And this is
not a tribute to fashion, 100% of
the content of the site (3 books, supplement and matlabprograms)
was written in 19842000 on Polish
Mazovia and IBM pc with
the help of the easily accessible text editor “ChiWriter"
without using paper and writing tools;
The foundation of classical
theoretical mechanics
is a finite set
of massive points without volume. In AMMG, an
infinite continuum set of
also nonbulk, but massless points
is taken as the foundation. This requires a deep rethinking of the
classical theory because it makes meaningless such concepts
as force, moment of force, particle, Newton's
axiomatics, everything that follows from it, etc .;
For
a productive understanding of the materials of the site
(the main content of which is 3 books, addition and Matlab
programs), it is desirable to agree with of the need
!!! of critical discussion and modification of the
following initial assertions of classical theoretical
mechanics;
From the point of view
adopted here (the first book of the AMMG site), the
phenomenological mechanics of an absolutely rigid body (ATT),
systems of solids and a continuous medium (fluid dynamics,
elasticity theory, etc.) are not independent branches of
mechanics. In the abovementioned conditions, it is proposed to
discuss the possibility of an axiomatic construction of the
mechanics of a point continuum. At the intuitive level, some
initial concepts are defined, independent consistent basic axioms
are formulated (including Galileo), necessary definitions are
introduced, etc. Using the above provisions, the primary (basic)
and the following statements are formulated in a form convenient
for of direct analytic transformations and computations on a
computer. All this (in aggregate) and is proposed to be
considered the beginnings of digital mechanics
of Galileo (Galileo Galilei, 15641642);
This emphasizes that we are
not talking about inventing some new mechanics. At the same time,
in the discussed variant of mechanics, there is practically
nothing from classical textbooks, monographs and articles from
this field of human knowledge, for example, (7), (16), (17),
(37), (38), (46) AMMG .
Sections of mechanics that are oriented to the study of a
continuum with specific physical properties: "droplet"
viscous liquids, gases (including ideal ones), guidingelastic
bodies, absolutely rigid bodies (ATT), their
systems, etc., should follow from general theory of
continuum mechanics.
We do not mean the simple
retelling of several classical monographs on
the indicated sections of mechanics from one point of view,
and also not about the fashionable filling of standard
classical mechanics with the concepts and terms of modern
mathematics [1,
47] (AMMG). Although the latter, to the extent of a fundamental
need in a small amount, is used in a variant that could
be considered modern applied mathematics in
the part concerned with mechanics;
The basis for these
constructions was the dissatisfaction with
many basic concepts of classical
mechanics:
Newton's axiomatics
(Philosophiae Naturalcais Principia Mathematica) is formulated
for the translational motion of really nonexistent points
with mass. During the 16th and 18th centuries,
the notions of a continuous environment, absolutely rigid bodies
(ATT) and rotational motion as scientific categories
either did not exist, or were not developedenough;
We need an
axiomatics formalizing the concept of a locally changeable
medium (2.12), as of a continuum point set of
points without mass, and for it  the concepts of
inertia (2.3), gravity (2.35)
and other "causes of motion and stoppage" of mechanical
systems, taking into account the goal set above. In this case, a
meaningful revision of the composition of the basic concepts and
laws of mechanics, as well as their mathematical formulations,
adequate to the classical ones, is necessary if the latter are
preserved. For example, among
the continuum mechanics axioms there is no second law
ofNewton, the concept of "particle", "force
applied to a point, to the body" and in general "force",
"moment of force", etc.;
It
is not possible to transfer the "rules of the game" to
the theory of mechanics of a locally variable continuum with
such concepts of the 16th19th centuries
as a finite set of massive points, a "particle", and
also a fragment of a locally variable medium, distinguished by
(physically nonexistent, pencildrawn) surfaces (a sphere, a
tetrahedron, a parallelepiped, and so on);
The
physically and mathematically unjustified use of certain
positions of the mechanics of an absolutely rigid body (the
method of sections, the principle of solidification, etc., which
form the basis of the classical mechanics of the continuous
medium [38]) as a "basis" for
studying the equilibrium and motion problems of the
abovementioned fragments of a locally variable
continuum environment;
With an
accuracy "to the contrary," the mechanics of
an absolutely rigid body and systems of
solids should be built on the basis of the
previously formulated mechanics of a locally varying
continuum with
additional requirements (for example, the constancy of the
distance between any two points of the medium);
The
vector algebra of geometric vectors (vector space V3), as the
mathematical basis of
classical mechanics of Newton, is absent in modern
mathematics (as
a section).
This discipline (after necessary refinement) is sometimes used as
one of the examples of
vector spaces in the theory of structures (as
a factor structure, whose elements (free vectors) are equivalence
classes  continuum sets of codirectional directed segments of
the same length);
In
AMMG, at the stage of constructing the theory, the
numeric ndimensional vector space Rn, (RnxRn) is used
as the main vector, ensuring the immediate use of modern
mathematical support in
solving theoretical and applied problems
of mechanics, bypassing the stage of tedious transformation of
mathematical models in the language of geometric vectors and
tensors;
This can be considered the
first step in creating elements of digital mechanics, oriented to
the active use of computers in solving practical problems from
the time they were formulated;
The
binary inner algebraic operation of a vector product on a
3dimensional vector space of geometric vectors V3 does not have
any standard property of the inner multiplication operation (it
is not associative, has no neutral element, has no
inverse element, is not commutative). It is not clear why,
when and in what sense this operation is called a
“multiplication
operation”;
For
any geometric vector from V3, a pair of geometric
vectors for
which this geometric vector is
a vector product either does not exist, or there
are infinitely many such
pairs. This fact is important, but in classical theoretical
mechanics it is practically never taken into account and is
not discussed anywhere;
In AMMG,
independent free vectors are used only for the formation of
fivedimensional sliding vectors (7.1), determined not
only by
the magnitude and direction, but also by
a directs of slip. A sliding vector generated
by a free vector is
not a vector, that is, it is not
an element of
some vector space (Proposition 7.2);
The main
objects in the considered variant of the
architecture of
mechanics are actively used the sums of sliding
vectors (5.166) (kinematic (5.68), kinetic (5.169),
(5171), dynamic screws (2.16),
(2.21), (5.189)) generatsed by free vectors fromV3 (7.3);
This is
ensured by the formulation of the laws of the mechanics of a
locally variable continuum in AMMG only
at the point for
the sliding vectors intersecting
in itthe densities of the
screws (kinematic,
kinetic and dynamic), for example (5.166). This is one of
the central provisions of the newly proposed architecture of
classical mechanics: the replacement of actions with
sixdimensional screws with
actions with their fivedimensional densities relative
to the Lebesgue measure. The exception is the mechanics
of ATT and
their systems (Chapters 5 and 6), where the theory of kinematic,
kinetic and dynamic sliding vectors and screws (Chapter 7) is
basic;
The
classical theory
of sliding vectors and screws, outlined in the
language of the algebra of geometric vectors, substantially
limits its use for theoretical research
and the solution of applied
problems, especially considering the two previous statements. In
the AMMG developed
a new theory in the language of modern
algebra with
the aim of transforming it into
an efficient, computationally convenient
and economical working apparatus (7.11), (7.12), (2.17)
(Chapter 7);
In classical
hydromechanics, after the identical decomposition of the
velocity vector of
an arbitrary point of the "particle" of the medium
into the
sum of three terms (the CauchyHelmholtz identity,
(3.6)), three principally new concepts
for the theory (attention
!!!, and only here, in the classical theory there are no other
ways to introduce these concepts):
 The
rate of translational movement of the particle;
 Rate of
"clean deformation" of the particle;

The vector of the angular velocity of
the particle (1 / 2rotv).
In the
classical theory of elasticity [38],
the Jacobi matrix
(the derivative of the displacement vector "z" along
the position vector of the point "y"  dz / dy) is
represented by the classical identity decomposition into
the sum of symmetric [dz / dy] and skewsymmetric
<dz / dy> matrices
(3.79 ).
For these matrices, two concepts that are fundamentally new
for this theory are
introduced:
 The
matrix of relative deformations of a particle;
 Matrix
of particle rotations provided
it "hardens".
These
concepts are elements of kinematic identities and
no relation to physics, movement of the real environment do
not have. "Particles" and their "turns" in a
locally variable continuum (not absolutely rigid bodies) do not
exist in principle. The theory itself in no
sense is the
kinematics of a continuous medium;
In AMMG,
a variant of the theory of the kinematics of a locally
varying continuum is proposed.
We introduce the notion (3x3) = of the matrix Dod, which
differs from the identity matrix E at the point "y" by
the infinitesimal matrix #
= DodE (2.3).
It is proved that the set of such matrices D00 = # + E in
the basis [ed], which accompanies the linear elastic
deformation of the medium, is the group (2.9).
The basis [ed] is such that [ed] = [eo] Dod.
It is suggested that the
result of the action of the operator Dod on the coordinate
column of
any vector xd in
the basis [ed] of the coordinate system Ed = (0d, [ed]) at
each moment of time t coincides
with the coordinate column of
this vector x0 in
the basis [eo] of the original coordinate system Eo
= (0o, [eo]), xd = xo at
the instant t
= 0 (2.13),
(2.14). The operators Dod are called kinematic
deformers (kdeformers)
of the
medium. The
matrices # approximately coincide with the matrices
ofthe relative deformations (3.27),
(3.10), (3.19);
There
are two physical reasons for changing the
indicated distances and
their velocities:
1. The
volume expansion of the medium (dilatation (3.42)),
whose geometric mathematical characteristic is
the divergence divu of displacement vectors
(u = z) or their velocities (u
= v), Op.3.3;
2.
The
complex shear (transvective (3.41))
relative displacement of points in the medium that occurs due to
the inequality of the cross partial derivatives of the
vectors and the
relative velocities v (or
the displacement z) along the coordinates of the position
vector of the point y (the
ith in the jth and jth in i ): dui
/ dyj and duj
/ dyi, i
= 3, 1, 2, j = 2, 3, 1. Such
a motion of environmental points can be conditionally
called stirring. Visually, it can be perceived in the
form of an unordered macroding, in particular, traditionally
unreasonably (at an intuitive level) called a vortex. On the
basis of the pseudokinematics of
the continuous CauchyHelmholt medium [16],
it is erroneously identified with the rotational motion. In the
case of elastic deformation, in the process
of transition to two
static states, these motions (stressed states) are likely to
be insignificant, but also exist. In the future, the term
"circulation" is used for the name of this movement.
The terms "turbulent" and "vortex" are not
used, as reminiscent of rotations, which are
essentially absent in
a locally variablecontinuum medium. The mathematical
characteristics of
this motion are the circulation coefficients of
the above vectors (3.93). They enter into the identity for
the Laplace operator
of these vectors (Lu = graddivu  T (Tu)) as the second term of T
(Tu), where, Tu = (dui
/ dyj
 duj
/ dyi,
i = 3, 1, 2, j = 2, 3, 1), T 
circulation operator, (Turbulence, twict 
circulation, vortex);
Note that
mathematical objects of
the form p2T (Tu), where p2 is
the new coefficient of
viscosity or stiffness, appear in the equations of motion of the
medium (as physical laws
of nature) in the process of their derivation on the basis of the
proposed axiomatics (4.110)
 (4.112) with (4.113) taken into account ) and have nothing to
do with the physically insoluble identity equations from
the classical kinematics of
the continuous (3.6) medium;
The concept
of circulations first
arose in the study of the formation of lift in the aerodynamics
of aircraft (1906,
NE Zhukovsky). This is the same as the above integral along
the boundary of the carrier profile from
the local velocity of
the medium. In a linear formulation, it is also determined
by a triple of functions, called the circulation
coefficients, and the profile area. Moreover, these
coefficients do not depend on
the indicated area. It was fairly believed that when the profile
boundary was "pulled" to the point, the lifting
force would tend to zero. The circulation coefficients that
do not depend on it remain the same, but they are now calculated
not at the boundary of the profile, but at the indicated
point. We emphasize once again that these coefficients have
nothing to do with any rotations of the medium. The triple
of the circulation coefficients with respect to the
displacement vectors z and
their velocities v (in
the traditional notation  rotu, where u = z or u = v),
in the algebraic sense, is the result of the action of
the circulation operator T on
these vectors and, in accordance with the classical theory,
determines vortex motion, recorded by numerical calculations and
experimentally. Other terms containing the circulation
coefficients Tu are
absent in the classical equations of motion of
a continuous medium;
In the
case of uncirculatsed (potentialer)
media motion, T(Tu) = 0 (the
integrand in
(3.91) is analytic),
the Laplace operators of
the coordinates of the vectors Lu coincide
with the coordinates of the divergence gradients of
these vectors graddivu. For an incompressible medium,
these terms become equal to zero and
the equations of motion of a viscous liquid are transformed
into equations of motion of an ideal fluid (ie, rheological
resistance in a viscous medium is absent). Hence it follows that
the potential (noncirculatsed) motion of an incompressible
viscous fluid does not exist. Perhaps both of these
phenomena do not exist in the real world. But if we assume that
the viscous medium is incompressible (which is often done in
solving practical problems), then its motion must be circularser;
The
relation between (3x3) = the stress matrix and
(3x3) = the matrix of relative deformations or
their velocities inclassical
mechanics is
represented by the linear equality (4.194) of the Genke
type [38]
(Genke, Heinrich Hencky, 18851951). There is no physical or
mathematical basis for
such a representation. An absolutely definite form
of the matrix function of
the matrix argument is
known, the matrix coefficients of which are
also calculated according to completely determined
rules (see "Theory of matrices", Gantmakher RF,
Nauka, 1967, [9], AMMG), which has nothing in common with
accepted in the classics of the appointment of their elements
as rheological coefficients;
The
timeproved experimental law of Hooke Hooke (16351703)
for deformable solids (stress proportional to relative
deformations) is generalized to the entire continuum medium
(which has already been done, for example, [46], AMMG): each of
the 9 voltages is the linear combination of all 9. relative
strains (3.18) or their velocities (3.17) under
the condition of reversibility and invariance of the
corresponding (9x9) = matrix of rheological
coefficients with respect to the rotation group in
R9 (Important (4.96)),
(4.22) , (4.30), (4.55), (4.105 )
... In
contrast to what has already been done in the case under
consideration (in the absence of symmetry (3x3) stress matrix),
the (9x9) equations of
the mechanical state of the medium are not equivalent (3x3)
= to Genk's relations (4.194);
A
separate conversation about gas
pressure.
In classical mechanics, this is the force per unit area of
the conditioned contact surfaces with
the environment (the faces of the . tetrahedron). In
the continuum environment,
there are neither forces and contact surfaces. At
the stage of creation and use of the phenomenological
theory, it is proposed to
call the above linear combination of all 9 actual relative
strains (3.18)
or their velocities (3.17)
that gas
pressure at
the point ofthe
continuum medium, ie, to determine a new physical quantity using
previously determined quantities. The physical content of
this concept is considered in molecular physics;
В общем
случае (при наличии вышеуказанной циркуляции и использовании
(9х9)=матричных уравнений механического состояния)
существует несколько
классов новых уравнений движения вязких
и упруго деформируемых сред, соответствующих разным
группам вышеуказанных
(9х9)=матриц реологических коэффициентов (Параграф
4.6.1);
They
include new ones (additional
to that contained in the identity Lu = graddivu  T(Tu)), which
depend of the circulation coefficients Tu of displacement
vectors and their velocities with
one more (previously unknown) rheological coefficient
p2 (viscosity,
stiffness), for example, p2T(Tu), (4.110)  (4.112) with
allowance for (4.113) in AMMG Note
that these terms appear in the process of their derivation using
the laws of physics in
theaxiomatics adopted here and have nothing to do with the
kinematics of the medium;
In the
uncirculatesd (potential) motion of media (flow, elastic
deformation), these terms disappear, (3x3)
matrixof deformations (velocities)
become symmetric (4.198).
Consequently, the entire classical theory of
a continuous medium (including the
theory of NavierStokes equations) is constructed for
the noncirculatsed (potential)
motion of the medium. It can be hoped that in the integration of
these new equations turbulent flows of a viscous fluid (
elastic deformation of the material) will be obtained and there
will be no difficulties in their investigation
and solution (similar
to the NavierStokes equations);
In AMMG (Chapter
4.7) it is shown that
the reason for the problems with integrating the
NavierStokes equations is
that these equations are not correct (4.181);
At the
stage of writing the monograph AMMG it
was assumed that information on these issues is sufficient for
further independent more detailed
consideration of
these issues. Discussion of
the topic with colleagues andreaders showed
that this is not so. A small attempt to correct the
state of affairs is made in the annotation to
the site. It is shown that under the
condition that
each pressure is
represented as a linear combination of all relative strain
rates, the mechanics of
a viscous fluid are not symmetric;
MECHANICS OF ABSOLUTELY
SOLID.
In this section, the next
(most significant) step is taken to construct
the fragments of digital mechanics, taking into
account the construction of the mechanics of an absolutely
rigid body and systems of solids. In
the classicalmechanics of ATT, the amount of
information on the theory of motion of an absolutely rigid
body (ATT) is insignificant. These are the simplest
statements, (Leonard Euler, 17071783) with one of
the 12 possible variants of the kinematics of
rotational motion and two forms of ATT dynamics
equations: the equations of translational motion of the
Newton mass center in the inertial coordinate
system and the Euler dynamic equations in
the central inertial coordinate system . The
rotation of the ATT with a fixed point is
considered. Particular cases of Euler, Poinsot, Lagrange
Lagrange (17361813) and Kovalevskaya (18501891)
are discussed in detail. A large number of applied works
by academician AY Ishlinsky on the study
of gyroscopes are known;
In the first
two monographs of the site, based on the proposed
axiomatics, an algebraic theory of the
kinematicsand dynamics of simple and complex
ATT motions, convenient for realizing on the
computer, was constructed in the general case (parallel
transfer and rotation). Then the kinematics of complex motion is
studied in the presence of preliminary (constructive)
parallel transfers and rotations, as well as in
the presence of simpleholonomic constraints (carried out
with the help of simple cylindrical, spherical and telescopic
joints). Using the above kinematics equations, ATT motion
equations are obtained (in the presence of simple holonomic
constraints);
In the classical kinematics
of ATT rotational motion, the parameters
of RodrigHamilton (William Rowan Hamilton,
18061865), Keyli Klein (Ceyler Arthur, 18211895,
Felix Chlein, 18491925) and quaternions are used.
In the standard presentation of the question, these are
various theories based on the use of vector algebra [37].
In the monograph a unified algebraic theory is
constructed on the basis of the theory of structures. It
turned out that there are not three, but one theory, up to
the isomorphic structures chosen for its construction;
Using
the concepts of inertial (5.170)
and kinetic (5.171) screws,
two new algebraic (6x1)forms of equations of
the general (translational and rotational) motion of ATT in
inertial (5.180), (5.182) and arbitrary coordinate
system (5.191)
 (5.193), (5.195) received .
In particular cases, these equations break up into two
inertially independent (3x1)vector equations: the
translational motion of the center of massthe Newton equation
(in the inertial coordinate system) (5.199) and the Euler
dynamic equations (in
the coupled coordinate system) (5.198). We emphasize that here
the first of these is no longer Newton's second law, but a
special case of the new general ATT motion equation (5.195)
(as a continuum set), not related to any
classical postulates
that do not have;
With the
use of the corresponding kinematics equations,
the ATT motion
equations are obtained in the presence of the
abovementioned simple holonomic constraints (5.206);
The
chapter concludes with the derivation of the algebraic (6x1)
= forms of ATT motion
equations in
the form A(q)q '' + (q, q ') q' = F(q, q ') carrying
dynamically unbalanced and asymmetric rotating
flywheels (without
feedback) (5.214) in the inertial external environment (for
example, of water) (5.217);
On
the basis of the developed theory, the ATT equations in the
Lagrange form of
the second kind in generalized coordinates (5.228),
(5.239) were obtained for the first time as a continuum
set, without using
the means and positions of analytical mechanics for
a finite set of physically nonexistent massive points. Matrix
ATT equationsare
obtained in the potential field (5.249), (5.266).
MECHANICS OF SYSTEMS OF
SOLID BODIES
Here the
digital ideology in mechanics is realized to a greater extent.
The basis for
constructing the mechanics of systems of solids
with tree structure is
for the first time fundamentally new for solving such problems,
the methodology of system analysis [39],
AMMG. The mechanics of arbitrary systems
of bodies (not having a tree structure) is obtained as a further
development of the indicated thorium using a method that is
ideologically similar to the reception of Lagrange multipliers.
The methodology can be viewed as a simple, visual
and
computationally economical
alternative
to symbolic computation [13], which allows direct computations,
bypassing the stage of obtaining bulky scalar equations;
The latter,
if necessary, is also possible, without the use of cumbersome
algorithms for
symbolic computation of derivatives and algorithms for reducing
such terms (using
only simple algorithms for symbolic multiplication of matrices).
The number of operations of
addition and
multiplication under
the proposed methodology for the standard most
computationally laborious test (a
sixlink kinematic chain with rotational kinematic pairs) is
about 20 times smaller
than when using symbolic computations for the Lagrange algorithm
of the second kind (the
second book of the site);
In the
development of kinematics, of the entire system of bodies (with
the structure of the tree!) Is divided into three subsets of
three hierarchical levels included in each other:
 Kinematic
pairs  the first level,
 Kinematic
chains (consisting of kinematic pairs)  the second level,
 The whole
system of bodies (consisting of kinematic chains)  the third
level;
For
each
kinematic pair, the matrix equations of kinematics
(kinematic
aggregates of the first level  in the terminology of system
analysis) are obtained. The form of the equations is ready for
immediate implementation on the PC. Constructive and functional
parallel transfers, rotations (including
gukelastic) and simple holonomic constraints are
taken into account (6.2);
The
equations of kinematics of pairs in
accordance with the tree graphs of
the system are "packed" into kinematic chain equations
(secondlevel aggregates) (5.71). From them the equations of
kinematics of
the whole system are constructed (aggregates of the third level),
(6.11);
Equations of
dynamics are constructed similarly. First, the above equations
for the motion of individual ATTs in (6x6) = matrix form are
formed taking into account holonomic links with neighboring
bodies (dynamic aggregates of the first level). (5.217). Then the
equations obtained using the first introduced matrices
(configurationalL (6.5)
and
structuralS (6.11))
are "packed" into the equations of motion of the entire
system (dynamic aggregates of the second level) (6.21). Equations
are subject to direct integration using any package that works
with matrices (for example, matlab);
These
matrices play a fundamental
role in
theory:
The
configuration
matrix L
forms a
mathematical model of kinematic constraints
in kinematic pairs (6.5), providing a massive use of
computationally economical recursions. The use of the
structural matrix S (6.11)
leads to a significant increase in the computational efficiency
of the equations, which increases with increasing number of
degrees of freedom. Both matrices have an absolutely
"transparent" structure, visually demonstrating the
contribution to the equations of motion of all geometric,
kinematic and inertial characteristics of the system. The block
structure of these matrices ensures
that in practice the above matrices play the fundamental role of
parallel parallelization of computations in the developed theory,
which greatly
increases
the computational efficiency;
The
structural matrix S, firstly, is used to exclude dynamic reaction
screws between
the bodies of
the system (6.18) (without the use of ideal coupling Lagrangian
work) and extracts the corresponding forces (6.19) from the
control screws and the friction in
the kinematics pairs. Secondly, it turned out that the final
matrix form of the equations of motion of the system of bodies
is formed
only with
the help of this matrix and
the constant blockdiagonal matrix of inertia of the system
(6.21). The blocks of
this matrix are the constants (6x6)
= the matrix of
inertia of the bodies of the system in the coordinate systems
(the Mises matrix) associated with them (5.172). In other words,
it was found out that for computer construction of
matrix equations of motion of a system of bodies in generalized
coordinates of
any dimension, it is sufficient to construct (calculate) only
two
matrices: a
constant matrix of inertia and
a structural matrix, which led to an additional increase in the
computational efficiency of the equations;
The
resulting matrix equations are clear. The inertial matrices
(of kinetic
energy for generalized accelerations A(q) and for generalized
velocities B(q,q*)) are represented as a product of matrices,
each of which determines the contribution to the dynamic inertia
screws (5.207),
(5.208) of various geometric and physical quantities (kinematics
in pairs, the kinematics of transitions from body to body, the
vectors of mobility axes in pairs, masses and moments of inertia
of bodies);
The main
effect is
achieved by the computer construction of numerical
(digital)
forms of the equations of motion of
the body system at each step of integration. From the very
beginning, the simplest (3x3) = matrices (6.2)  (6.4) that
determine the kinematics and inertia of solids (6.11) (aggregates
of the second level), the process is completed by the formation
of numerical forms of generalized forces,
inertia matrices and
a structural matrix using
a treelike of grafa system;
Finally, we
obtain readytouse computer
integration procedures (for example, MATLAB) for the equations of
motion of the system in the form A(q)q ** + B(q,q*)q* = Q(q, q*)
with the
matrices already calculated and the vector Q(q,q*),
(6.21) (timeconsuming operations for computing matrices and
generalized forces are practically not used). If necessary,
analytic forms of
the equations of motion of systems of bodies are constructed
using the
same scheme using economical matrix modules of
systems of analytical computations [13];
The
equations of motion of systems of gukelastic solids
have the
form (6.27). In paragraphs 6.2.3 to 6.2.6 methods of optimizing
the
algorithms for calculating the inertia matrices A(q), B(q,q*)
of the
equations of motion of systems of solids are given,
simultaneoused solving
the problems of control over
the correctness of their calculation;
At all stages of the
construction of the theory, a large number of examples of its
practical implementation are considered (gyroscopes on a mobile
base, manipulators, walking apparatuses, multitier stands  the
third monograph of the site, etc.).
COMPUTER TECHNOLOGIES OF
RESEARCH OF MULTIPURPOSE MECHATRON SYSTEMS
In the last
monograph of
the site, based on the above theory, from a single point of view
(in coauthorship with an expert in electrical engineering and
electronics, Ph.D., assistant professor of automatic control of
Spb GETU (LETI), director of the Laboratory of Industrial
Electronics Infineon Technologies Sergey Viktorovich Gavrilov)
considered issues at the junction of several areas of human
knowledge: the newly constructed mechanics of systems of solids,
electrical engineering, electronics, management theory and
computer science. For the first time the approach to the study of
mechatronic systems is
presented not as a conglomerate of independent abovementioned
disciplines. This is a unified statement of the issues at the
junction of these disciplines on
the basis of modern algebra in numerical form, ready for direct
use on a computer without any
preliminary preparatory work;
For the
first time, methods have been developed for testing the matrix
equations of motion of solids systems (for adequacy to the laws
of mechanics) and the program for calculating them
(several different exact algorithms for computing inertia
matrices) (paragraphs 6.23 to
6.24). For comparison control, scalar
equations of flat motion of
the stand are obtained using the developed theory
and Lagrange
equations of the second kind;
The dynamics
of the stand, of represented by a springloaded massive
base
carrying arbitrarily installed and
oriented dynamically unbalanced flywheels, whose rotation is
performed by controlled electric motors, is studied: the
identification of the system has been formulated
and proved;
necessary and sufficient conditions for selfsynchronization of
rotation of the rotors and ranges of its speeds; One rotor
due to
active rotation of the other rotor.
The effects of temporary
deceleration with subsequent restoration of the speed of passive
rotation of one of the rotors are investigated. It is shown that
all these effects are realized due to the equations of motion B
(q, q *) q * quadratic in velocities of inertial terms and only
for a certain combination of kinematic, geometric and inertial
characteristics of the stand. With an arbitrary combination of
these characteristics, these effects are absent;
The site contains two Matlab
programs (computational  STI.M and integratingSTIR.M), allowing
to solve a wide class of problems of the dynamics of the
abovementioned stand, including tracing all the abovementioned
effects. In this case, it is necessary to take into account that
the identification of the system of equations was carried out for
a specific "live" stand.
QUESTIONS OF GEOMETRICAL
OPTICS OF RADIOTELESCOPES WITH HYPERBOLIC AND ELLIPTIC
CONTROLLECTORS
This additional section of
the site contains the results of officially unpublished studies
carried out at the request of a colleague whose information may
be of interest to the creators of antener installations for
various purposes (television, radar, astronautical, etc.);
Using the methods of
classical analytical geometry, computeroriented methods for
mathematical modeling of the exact position of light rays (of as
geometrical objects) are developed for primary reflection from a
parabolic mirror and secondary reflection from hyperbolic or
elliptical counterreflectors. The possibility of the position of
the incident rectilinear wave fronts orthogonal and inclined to
the parabola axis is taken into account. For both
counterreflectors, computational programs with the use of the
MATLAB package (rtel.m, rtg.m) have been developed, numerous
studies have been performed, the results of which are presented
in the corresponding graphs;

The page was
last edited in September 2018.
CONTACT INFORMATIONE
Email: mechanics
konoplev@yandex.ru Tel.:
8 (812) 783 4278 Mob.: +7 (921) 351 0175
Postal address:
Konoplev V.A., pr. Stachek 671 66, 198096 St.Petersburg, Russia
