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
Preface
to
the web Algebraic
methods in
Additional
chapters:

Abstract 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 NavierStokes 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 CauchyHelmholtz 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 skewsymmetric 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), 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 9dimensional 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 9dimensional column of voltages made up of the columns of the nonsymmetric (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 9stresses Tij with the elements (dvi/drj) = (vij) of the above nonsymmetric (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 skewsymmetric 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, T12T21 = p3cv3, T31T13 = p3cv2, T23T32 = 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. Nonzero distributed tangential stresses and their moments confirm the nonsymmetry of (3x3) = stress matrix (tensor) and, consequently, assert (according to LGGoitsanskii) the nonsymmetry of the fluid mechanics of the viscous fluid; 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 NavierStokes 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 NavierStokes 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 NavierStokes (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) 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: rov* = rog  gradp  mrotrotv = rog  gradp + (p2  m)T(T(v)); (13) New computeroriented, computationally efficient mechanics of rigid body (RB) had been built; When building the mechanics of solid bodies systems, two fundamental matrices were obtained: Lconfigurational matrix and Sstructural matrix; 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; Email: mechanicskonoplev@yandex.ru


