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



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


MATLAB – programs of
computer modeling of
geometrical optics
questions

rtel.m    rtg.m

 

 

Abstract

The main goal of the site is to create one of the possible variants of the beginnings of digital mechanics. This refers to the construction of a new theory focused on the economical massive use of computer technics and it-technologies in solving theoretical and practical problems (including the new version of computer algebra [13] AMMG (the first of the books [13]);

  The foundation of classical theoretical mechanics is a finite set of massive points without volume. In AMMG, this the foundation is replaced by an infinite continual set of points without volume, but  without mass. In this case, a deep rethinking of the classical theory is required, since it makes meaningless notions such as force, moment of force, particles, Newton's axiomatics, everything that follows from it, and so on;

 On this foundation, the house of theoretical mechanics of the new architecture was built, in which there is practically nothing that exists in [7, 16, 17, 37, 38, 46 ...] and their foreign counterparts. However, after wandering in this house with due diligence (not "from corner to corner"),  can rethink many of the usual concepts and  statements;

 Below is a simplified version of the theory, traditionally called "aerogasodynamics" in the part relating to the motion of a viscous medium, as well as brief information about other branches of mechanics (absolutely rigid body of ATT, systems of solids, mechatronic systems, etc.). In all cases, priority is given to the convenient and economical use of computers in solving theoretical and applied problems;

 Professor Loitering L.G. in the book "Mechanics of liquid and gas." - St. Petersburg: Science. 1978. - 532 pp., proving the symmetry (6x6) = stress matrix (tensor), noted that in the case of distributed moments in the liquid, the indicated matrix is not symmetric. The mechanics of the continuous medium in this case he called "not symmetrical mechanics";

 In AMMG it is shown that under certain conditions the mechanics of a viscous fluid are not symmetric. These conditions "permeate" the entire theoretical hydromechanics of a viscous fluid, from the first pages of classical textbooks to the Navier-Stokes equations (NS) inclusive, which was one of the reasons for the name of the site. All statements, of course, require a comprehensive experimental and computational verification. The purpose of this work, among others, is to stop the attention of interested readers on this fact;

 The Cauchy-Helmholtz formula AMMG [16], (3.6) (the expansion of the deformation rate of a liquid into two components: pure deformation and elementary rotation) is a geometric (kinematic) identity. The "right" part of the equation is another notation for "left" and can not be the basis for introducing new physical objects into theoretical studies. In particular, the symmetric matrix of pure relative deformation rates and the vector of instantaneous angular velocity 2w = rotv in classical hydromechanics appear from nowhere. Particles do not really exist, and the vector rotv has nothing to do with the rotation of these particles;

 In AMMG, in the part concerning the continuous environment, two new central concepts have been introduced, within the framework of the ideology under discussion:

 - First, it is an asymmetric (3x3) = Jacob matrix

 dv/dr = (dvi/drj) = (vij), (i, j = 1, 2, 3) -                                                                                  (1)

 (3x3) = the matrix (tensor) of the relative deformation velocities (but not the symmetric part [dv/dr] of its expansion into the sum of the symmetric and skew-symmetric terms dv/dr = [dv/dr] + <dv/dr>), as is commonly assumed in the classical construction of fluid mechanics of a viscous fluid;

 Secondly, it is the circulation of the velocity vector at any point of the fluid (3.91). This fundamentally distinguishes the mathematical object under consideration from the classical circulation of N.E. Zhukovsky, defined on the boundary of the streamlined profile and generating the lift force;

 Let us calculate the circulation of the linear representation of the velocity vector dvz = vr + dv/drzdz along the boundaries of r + z circles with the normals in the form of orthovs ei, (i = 1 ,2, 3) of the orthonormal Cartesian coordinate system starting at any point r of the fluid. We obtain a triple of functions cvi independent of the areas of these circles, each of which is the difference in the rates of change (j, k) = the coordinates of the velocity vector of the orthogonal ei with the change in the cross coordinates of the position vector of the point (jth in the kth and vice versa) (3.92, 3.93)

 cv1 = v32 - v23, cv2 = v13 - v31, cv3 = v21 v12-                                                                                      (2)

 where vjk = dvj/drk is the partial derivative of the jth - coordinate of the velocity vector v = (v1, v2, v3) along the kth - coordinate of the position vector of the point r = (r1, r2, r3). Vector 

 Cv = (cv1, cv2, cv3)-                                                                                                                   (3)

 completely defining the circulation of a viscous fluid moving with velocity v, in AMMG is called the circulation vector, and the of fluid flow is circulated;

 A new concept "circulation operator T" is introduced, such that T(v) = (cv1, cv2, cv3). Formally, T(v) = Cv = rotv, but the geometric and physical meaning of these mathematical objects are fundamentally different. It is assumed in AMMG that the objectively existing, experimentally recorded circulation (3.92) is a geometric (kinematic) mathematical model of the vortex formation (turbulence) of a viscous liquid of different intensity (greater in the boundary layer during flow along a solid surface, in the vicinity of the confluence boundary of two flows with different velocities, on a free surface such as a cavern or a trace behind a glider, etc.);

 A direct relationship between fluid flow is not circulating T(v) and the matrix dv/dr is established: if the matrix coincides with its symmetric summand

 dv/dr = [dv/dr], <dv/dr> = 0-                                                                                                       (4)

 then the fluid flow is uncirculated (potentialed),

 T(v) = Cv = rotv = 0;

 The following in the provision of the topic in AMMG is the proposal: each of the 9 voltages in a viscous liquid Tij is represented by a linear combination of all 9 relative strain rates vij = dvi/drj of a liquid with a persymmetric (symmetric relative to both principal diagonals) (9x9) = a rheological of matrix coefficients M (rt). The matrix meets two requirements: firstly, the matrix must not be special, that is, it ensures the restoration of relative strain rates at known stresses (4.26), and secondly, it must be an invariant of the group (9x9) = rotations (4.27), as an element of mechanics Galileo. The resulting matrix equation is called the equation of the mechanical state of the liquid at the point r (4.30).

 (T) = - (p) + A + M (V)-                                                                                                                  (5)

 where (p) = pcol (1,0,0,0,1,0,0,0,1,1), p is the Pascal pressure, A = lcol (1,0,0,0,1,0,0,0,1), l = coefficient of viscosity of the first type (lambda), M - the above (9x9) = matrix of viscosity coefficients of the second type with a coupling (4.96), (V) = (v11, v21, v31, v12, ..., v13, v23, v33) is the 9-dimensional column composed of the columns of the asymmetric (3x3) = matrix of the relative velocities of the fluid deformation (vij = dvi/drj), ( 4.2), (T) = (T11, T12, T13, ..., T32, T33) is the 9-dimensional column of voltages made up of the columns of the non-symmetric (3x3) = stress matrix (tensor) (4.3). In AMMG several liquids with such matrices are considered (Ch. 4.2);

 Consider a fluid with the equation of mechanical state (4.105) and the constraint (4.109), which ensures the fulfillment of the two above requirements for the matrix M(r, t). Rewriting the matrix equation in the form of nine scalar linear equations, we obtain a direct relationship of the 9-stresses Tij with the elements (dvi/drj) = (vij) of the above non-symmetric (3x3) = matrix (tensor) of the relative rates of deformation of a viscous liquid in the classical form Tij = mdvi/drj (AMMG [17]), where m is the coefficient of friction (of viscosity). We note once again that the matrix dvi/drj is not symmetric (contains its skew-symmetric term <dv/dr>, declared in the classical mechanics of a continuous medium by the matrix of rotation (rotations), and therefore, in the authors' opinion, not involved in creating dissipative stresses);By a simple subtraction of Tij -Tji, we obtain three equalities that determine the direct connection of tangential stresses with the coordinates of the circulation vector (3)

 T (v) = Cv = (cv1, cv2, cm3) = rotv,

 T12-T21 = p3cv3, T31-T13 = p3cv2, T23-T32 = p3cv1-                                                             (6)

 where p3 is the experimental coefficient of viscosity. Objectively existing circulation of the velocity vector (vortex formation, turbulence) is a mechanism of energy loss of liquid at the points of its existence (boundary layer, etc.), the costs of which are proportional to the work of stresses. Non-zero distributed tangential stresses and their moments confirm the non-symmetry of (3x3) = stress matrix (tensor) and, consequently, assert (according to LGGoitsanskii) the non-symmetry of the fluid mechanics of the viscous fluid;

 The dilatational stresses (tensile, compression) in the direction of the unit vectors of the coordinate system depend on the elements of the main diagonal of the asymmetric matrix of the relative velocities of the fluid deformation

 Tii = p2divv + p3vii, divv = (v11 + v22 + v33)-                                                                           (7)

 where p2 is the second experimental coefficient of viscosity;

 The results discussed have a geometric (kinematic) nature. Of no less interest is the contribution of circulation to the dynamics of a viscous fluid. Substituting the equation of the mechanical state (4.105), (5), taking into account the equality (2.44), into the equations of motion (2.65), we obtain equations for the dynamics of a viscous fluid (4.118) - (4.120). Assuming that the viscosity coefficients do not depend on the point r in the initial inertial coordinate system, the obtained equations of motion can be rewritten in a simplified vector form

 rov* = - gradp + rog + p2T(T(v)) + (l + p2)L(v)-                                                                         (8)

 where (ro) is the density of the liquid, p is the Pascal pressure, g is the acceleration of gravity, l is the classical viscosity coefficient (lambda), L(v) is the Laplace operator, T(.), p2 is the above circulation operator and the viscosity coefficient. Thus, in addition to classical equations, a new term for the density of dissipative forces (viscous friction) p2T(T(v)), which is the product of the viscosity coefficient p2 on added to the equation of motion of a viscous fluid. It this is the result of the action of the circulation operator T(.) on the circulation vector T(v). In the classical notation, the result can be rewritten as p2T (T (v)) = p2T (rotv) = p2rotrotv;

 The kinematic equations of the Navier-Stokes mechanical state () have the form

 Tns = (- p + ldivv)E + 2m [dv/dr]-                                                                                              (9)

 where Tns is the symmetric (3x3) = stress matrix, l, m are the above traditional viscosity coefficients (lambda and mu), E is the unit (3x3) = matrix, [dv/dr] is the symmetric matrix of relative strain rates.Simplified (l, m - constant) equations of Navier-Stokes motion have the form

 rov* = rog - gradp + (l + m)graddivv + mL(v)-                                                                       (10)

 In deriving these equations, the symmetri of the asymmetric matrix of relative deformation rates dv/dr = [dv/dr] + <dv/dr>, adopted in the AMMG, is used as a matrix of relative velocities deformation of a viscous liquid, vector of circulations T(v) (5), (6) was not used. The question arises: whence, in the numerical solution of the Navier-Stokes (NS) equations, a vortex formation (turbulence), confirmed by experiment, appears if there are no dissipative terms of viscous friction T(T(v)) in them?. Possible that the derivation of the equations (NS) is not complete? If the Laplacian transformation is continued using the identity L(v) = graddivv - rotrotv known in the vector analysis, we obtain the equations (HC) in the form

 rov* = rog – gradp + (l + 2m)graddivv – mrotrotv =

 = rog – gradp + (l + 2m)graddivv — mT((Tv))-                                                                      (11)

 where mrotrotv = mT(T(v)) is the density of the dissipative forces of viscous friction determined by the circulation vector T(v), (6), ie, in full accordance with assertions (4), the dissipative force of the molecular friction of a viscous fluid is determined by the asymmetric matrix of relative deformation rates dv/dr = [dv/dr] + <dv/dr>: divv and T(v) = rotv;

 If the equations of motion of a viscous fluid (8) obtained in AMMG are transformed using the above identity L(v) = graddivv - rotrotv, we obtain

 rov* = - gradp + rog + (l + p2 + m)graddivv + (p2 - m)T(T(v));                                            (12)

 These equations of motion for a viscous fluid differ from equations (HC) in that:

 The equations from AMMG (12) are obtained using the equations of the mechanical state of a liquid in the form (5),equations (HC) - using equations (9), not equivalent to equations (5)

2. In deriving the AMMG equations (12), the asymmetric (3x3) = matrix of relative fluid deformation rates dv/dr is used, in deriving of the equations (HC), (10) - the symmetric part [dv/dr] of this matrix dv/dr is use

3.    By virtue of paragraph 2. (10) are the equations are for the potential motion of a viscous fluid, since the condition dv/dr = [dv/dr], <dv/dr> = 0 and the fluid flow is uncirculated (potentially), T(v) = 0, EQUIVALENT, (4). The turbulence registered in the numerical integration and experimental studies of equations (10) is the result of the presence in the Laplacian L(v) of the term rotrotv = T(T(v)), determined by the circulation T(v), by virtue of the identity L(v) = graddivv – rotrotv;

The equations of motion of an incompressible viscous fluid obtained in AMMG using the equations of the mechanical state (5) have the form

 rov* = rog - gradp - mrotrotv = rog - gradp + (p2 - m)T(T(v));                                              (13)

 I would like to understand how the above information can influence the solution of one of the Hilbert problems formulated by the Clay Mathematical Institute (Cambridge, Massachusetts, USA) on May 25, 2000 in the field of investigation and integration of the equations of motion of a viscous incompressible Navier-Stokes fluid (NS);

 In AMMG, for the investigation of a viscous fluid and elastic material, the same mathematical apparatus was used for an abstract vector u such that u = v (v is the velocity vector) in the final results if the medium is a viscous liquid, u = z (z is the displacement vector), if the medium is elastic material. When determining the displacement vector z (3.14), (3.15) (and there are no others), the use of this theory of elastic medium is most likely doubtful because of the lack of physically reasonable meaning in the mathematical concepts dz/dr, divz, rotz, etc.;

 New computer-oriented, computationally efficient mechanics of rigid body (RB) had been built;

 The algebraic kinematics of simple free, bound and complex RB motions is constructed that participates in any number of relative motions in the presence of constructive parallel translations, rotations and motion constraints with respect to some generalized coordinates;

 A new (6x6) = matrix equation of kinematics is obtained that describes the generalized velocities and quasi shortness of RB in simple bound motion;

 A new (6xdimq) = matrix equation of the kinematics of complex motion is obtained, where is the generalized coordinates of the RB. Equations are used to construct the kinematics of systems of solids with tree structure;

 Algebraic variants of kinematics of the simple rotation of a rigid body in terms of the representation theory of the rigid body rotation group are constructed with help Rodrigue-Hamilton , Cayley-Klein groups, and body of quaternions are constructed. It is shown that they are indistinguishable before the accuracy up to isomorphism;

 At the establishment of fundamentally new (66)=equations of rigid body motion in an arbitrary orthonormal body coordinate system, Newton's equations and Euler's dynamic equations are not used;

 When selected major central axes of inertia, they are converted into the above-mentioned classical equations, but these are no longer the laws of nature, but the particular cases of the new (66)=matrix equation of rigid body motion;

 Equations of motion of a rigid body with simplesty non-holonomic constraints as well as of solid body carrying mobile massive parts (without backward communications,) in an inertial external medium with a potential flow are studied;

 These equations are used to create aggregative dynamics of a system of solid bodies with a tree structure based on system analysis as a first-level aggregations;

 Using the established rigid body equations derived on the system analysis, a fundamentally new computer-friendly, computationally efficient mechanics of solids body systems has been developed (the second monograph of the site);

 First-level kinematic aggregates (matrix above-mentioned differential equations of kinematics of kinematic pairs) are constructed first, then from them, in accordance with the system graph, second level aggregates are constructed (differential equations of kinematics of kinematic pairs). From these aggregates, also according to the system graph, are constructed (6ndimq)=matrix differential equations of kinematics of the entire bodies system with the tree structure (n is the number of bodies in the system, q is the generalised coordinates of the system);

 Similar constructions are performed when constructing dynamics of a solid bodies system with a tree structure. In the capacity of the first-level aggregates, the above-mentioned (66)=matrix equations of rigid body motion are used. In the capacity of the second-level aggregates, equations of kinematic chains motion (based on the elasticity of a rigid body) are used. In the capacity of the third-level aggregates, (dimqdimq)=matrix differential equation of the movement of the entire bodies system are used;

 When building the mechanics of solid bodies systems, two fundamental matrices were obtained:

L-configurational matrix and S-structural matrix;

 The configurational (6nx6n) = matrix L contains exhaustive information about all the kinematic characteristics of the system, the structural (dimqx6n) = matrix S eliminates the dynamic screws of internal reactions in the kinematic pairs of the system without the work of ideal bonds and allocates control forces from the dynamical system control screws;

 Constructions of models based on the system analysis provides a high computational efficiency compared to the use of Lagrange equations of the second kind (approximately twentyfold for a standard test six-unit kinematic chain with rotational kinematic pairs);

 The additional efficiency of computation is achieved by calculating process paralleling on multiprocessor computers and use of recursions, enabled by this technology practically unlimited;

 The resulting equations are subject to immediate integration, bypassing the stage of the cumbersome construction of scalar equations. At each step of integration (e.g. using the matlab procedure [t,x]=ode45(,sti, [t0 tf], x0), at first! (33)- matrix blocks are calculated, then numerical (66) cubes are constructed from the above-mentioned equations of each rigid body motion, from which numeric inertial (dimqdimq)=matrices of the A(q), B(q,q*) system are later constructed. Finally, of external actions are calculated (see matlab files sti.m,stir.m  in the site menu);

 Construction of scalar equations of motion (if necessary) is also possible with the use of efficient matrix modules of analytical computing systems;

 In the third monograph (see the menu of the site), a computationally efficient computer technology for research on the dynamics of mechatronic systems at the confluence of several areas of human knowledges has been developed (newly constructed mechanics of solids body systems, of electrical engineering, ofelectronics, management theory, informatics). A large number of practical examples were examined;

 In particular, there has been research on a mechatronic system, presented by a massive spring-loaded base, where two arbitrarily-placed and oriented massive flywheels powered by controlled electric motors were installed;

 A methodology has been developed to test the accuracy of the inertia matrix calculations A(q), B(q,q*) in constructed motion equations. The test of the correctness of the equations themselves is verified by their adequacy to classical laws of physics and by comparison with similar equations obtained using the Lagrange algorithm of the second kind;

 The last edit of this web page was completed in 2018, taking into account discussions with readers and colleagues.

 

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