New architecture of classical mechanics
Vladimir Andreevich Konoplev

Algebraic methods in Galilean mechanics. Agregative mechanics of rigid body_systems.
Computer technologies in research of multichain mechatron systems.

switch to russian

Abstract to the web

About author Konoplev V.A.

Preface to the  web  

About Navie-Stoks equations


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
radio-telescope with
hyperbolical and elliptical
Konoplev V.A.
( PDF rus) 

MATLAB – programs of
computer modeling of
geometrical optics

rtel.m    rtg.m




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 Matlab-programs, 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 matlab-programs) was written in 1984-2000 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 non-bulk, 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 above-mentioned 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, 1564-1642);

  • 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), guiding-elastic 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 non-existent 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 16th-19th centuries as a finite set of massive points, a "particle", and also a fragment of a locally variable medium, distinguished by (physically nonexistent, pencil-drawn) 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 above-mentioned 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 co-directional directed segments of the same length);

  • In AMMG, at the stage of constructing the theory, the numeric n-dimensional 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 3-dimensional 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 five-dimensional 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 it-the 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 six-dimensional screws with actions with their five-dimensional 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 Cauchy-Helmholtz 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 skew-symmetric <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 # = Dod-E (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 (k-deformers) 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 i-th 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 pseudo-kinematics of the continuous Cauchy-Helmholt 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, 1885-1951). 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 time-proved experimental law of Hooke Hooke (1635-1703) 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 Navier-Stokes 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 Navier-Stokes equations);

  • In AMMG (Chapter 4.7) it is shown that the reason for the problems with integrating the Navier-Stokes 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;


  • 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, 1707-1783) 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 (1736-1813) and Kovalevskaya (1850-1891) 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 Rodrig-Hamilton (William Rowan Hamilton, 1806-1865), Keyli- Klein (Ceyler Arthur, 1821-1895, Felix Chlein, 1849-1925) 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 mass-the 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 above-mentioned 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).


  • 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 six-link 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 guk-elastic) 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 (second-level 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 (configurational-L (6.5) and structural-S (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 block-diagonal 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 tree-like of grafa system;

  • Finally, we obtain ready-to-use 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) (time-consuming 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 guk-elastic 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, multi-tier stands - the third monograph of the site, etc.).


  • In the last monograph of the site, based on the above theory, from a single point of view (in co-authorship 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 above-mentioned 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 spring-loaded 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 self-synchronization 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 integrating-STIR.M), allowing to solve a wide class of problems of the dynamics of the above-mentioned stand, including tracing all the above-mentioned 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.


  • 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, computer-oriented 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.


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