Generalized cyclic displacements : PMM vol. 37, n6, 1973, pp. 1135–1137

We consider the generalized cyclic displacements of holonomic mechanical systems with a finite number of degrees of freedom, and their application to integration of the equations of motion.N.G.Chetaev in [1] turned his attention to the formulation of problems dealing with general properties of mechanical systems and connected with the groups of transformations which leave the basic mechanical functions invariant. It was he who introduced [2] the concept of cyclic displacement of a mechanical system with smooth holonomic constraints. This concept was enlarged in [3] in the course of considering a particular case of motion of a mechanical system with three degrees of freedom.     

On the membrane state of multiply-connected convex shells : PMM vol. 37, n6, 1973, pp. 1141–1145

Conditions for the realization of the membrane state of equilibrium of multiplyconnected shells of positive Gaussian curvature subjected to surface and edge forces, are investigated; the concepts of correctness and stability of the membrane states are introduced. The terminology and notation correspond to that used in [1, 2].     

On the generalization of the Bonnet theorem : PMM vol. 37, n6, 1973, pp. 1138–1141

The well-known Bonnet theorem [1] is generalized to the case of motion of material points of variable mass moving under the action of a quasi-positional system of forces, i.e. of a system where each force is a function only of those parameters which determine the position of the point on the trajectory.     

Investigation of the free vibrations of noncircular cylindrical shells : PMM vol. 37, n6, 1973, pp. 1125–1134

An asymptotic method of integrating the free vibrations equations of a noncircular cylindrical shell, freely supported along the curvilinear edges, is proposed. The qualitative singularities of the vibrations associated with the fact that the shell is essentially noncircular are clarified. Numerical results are presented for the free vibrations frequencies and modes of a box shell which are compared to the results of the asymptotic analysis. It is supposed that the variability of the stress and strain states is great.     

On a method of solving dual integral equations : PMM vol. 37, n6, 1973, pp. 1087–1097

We expound a method of reducing a class of dual integral equations which find important practical application to infinite algebraic systems of the first kind. The latter system can be reduced to the systems of the second kind by exact inversion of the principal singular part, and the second kind systems can be solved using the method of consecutive approximations [1–6], The dual integral equations generated by the Kontorovich-Lebedev and Mehler-Fock integral transforms are considered as examples as well as the problems of torsion of a truncated elastic sphere by a punch and that of a circular crack in an elastic space.     

On stability of three-dimensional periodic motions in hydrodynamics : PMM vol. 37, n6, 1973, pp. 1044–1048

We obtain exact conditions for the stability of periodic motions. We show that the conditions found in [1] are necessary and sufficient, but they are only applicable to motions not dependent on time. The conditions given in [2] are applicable in the general case but are only sufficient (necessary) conditions of instability (stability). We consider the dependence of stationary motions on parameters.     

On some variational principles in mechanics of continuous media : PMM vol. 37, n6, 1973, pp. 963–973

Some problems of the analytical mechanics of continuous media are considered. The kinematc constraints, restricting the motion of elements of a continuous medium are separated into internal and external constraints; the internal surface stresses in a continuous medium are treated as reactive forces of the internal constraints. Since the work of these latter on possible displacements of elements of the continuous medium is not zero in the general case, then the internal constraints in a continuous medium should be referred to the category of nonideal constraints. The general equation of the dynamics of a continuous medium expressing the d'Alembert-Lagrange variational principle and including all the dynamic laws is examined. The work of the internal surface stresses in this equation can be given by using the first and second laws of thermodynamics, whereupon the general equation of the dynamics of continuous media can be represented in two other forms. Furthermore, an extension of the Gauss and Chetaev principles to continuous media is given.     

Self-similar problem of Cauchy-Poisson waves at an inclined shore : PMM vol. 37, n6, 1973, pp. 1035–1039

We consider a two-dimensional problem concerning Cauchy-Poisson waves at an inclined shore in the case of an initial disturbance concentrated near the shore edge. We study the behavior of the solution near the shore and at large distances from it.Numerous investigations, devoted to the study of standing and progressive waves on an inclined shore, are described in [1]. A two-dimensional problem concerning nonstationary waves on a shore with an angle of inclination γ = π/2n, where n is an integer, was analyzed in [2, 3]. We consider below a case in which the angle of inclination is commensurable with λ/2, subject to the condition that the initial disturbance is concentrated in the vicinity of the shore edge, so that the problem may be considered self-similar.     

Certain qualitative investigation methods for dynamic systems connected with field rotation : PMM vol. 37, n6, 1973, pp. 984–989

We show that the inevitability of realizing bifurcations connected with a double (and triple) limit cycle or with a separatrix loop can, in some cases, be detected from the global evaluation of trajectory behavior under parameter variations by reckoning the sign of the saddle term, and we turn our attention to new possibilities of tracing the bifurcations arising from the use of a monotonic field rotation. The methods of bifurcation theory [1] are widely used in problems of mechanics. However, in the general case, the problem of investigating all possible bifurcations is difficult and regular methods for solving it do not exist. There are no criteria locally connected with the points of the phase space or of the parameter space, which stipulate the actual realizability in a concrete dynamic system of bifurcations connected with a separatrix loop. A similar situation exists for bifurcations connected with the arising of limit cycles from the condensation of trajectories, because we usually know neither the equations of the limit cycle nor the parameter values under which it arises. In a number of the methods used in bifurcation theory an important role is played by the local rotation of the field in a neighborhood of the singular trajectories of the system [1, 2]. In a number of cases the carrying over of this idea to the whole phase space and to the parameter space in the large (realizable in the presence of specific singularities of the system being investigated) permits us not only to trace all bifurcations possible in the system but also to predetermine the disposition of the bifurcation curves or surfaces.     

Application of the Bubnov-Galerkin procedure to the problem of searching for selfoscillations : PMM vol. 37, n6, 1973, pp. 1015–1019

We propose the use of the Bubnov-Galerkin procedure to the search for self-oscillations. We establish the existence and the convergence of the approximations. In the basic case we have obtained the asymptotics of the rate of convergence. In [1] it was shown, on the basis of the results in [2], how we can construct finite-dimensional approximations to the periodic solutions of autonomous systems. Below we have pointed out another approach to solving the approximation problem, based on the parameter functionalization method proposed in [3].     

Continual mechanics of monodisperse suspensions. rheological equations of state for suspensions of moderate concentration : PMM vol. 37, n6, 1973, pp. 1059–1077

Rheological relationships linking mean and moment stresses and, also, the force and moment of interphase reaction in a macroscopic flow of small solid sphere suspension with the kinematic characteristics of the flow are derived. This makes it possible to close the system of equations of suspension hydrodynamics. Coefficients of viscosity and of moment viscosity of a suspension are obtained and calculated.The equations of conservation of mass, momentum and moment of momentum of suspension and of its phases, considered (from the macroscopic point of view) to be coexistent continuous media, were formulated in a general form in [1]. These equations contain unknown vectors and tensors which define the interaction between the considered continuous media and, also, stresses and moment stresses appearing when these are in motion. To close the equations of conservation it is necessary to express all these quantities in terms of unknown variables of these equations (mean concentration of suspension, pressure in the fluid phases, and phase velocities). This problem is the second of the fundamental problems of hydromechanics of suspensions indicated in [1].Here this problem is solved with the use of a kind of self-consistent field theory, which is essentially an extension and generalization of methods developed in [2 – 7]. Expressions for all of the quantities mentioned above are derived. They can be considered to be rheological equations of state for suspensions. Expressions for the various coefficients of these equations and their dependence on parameters of phases and on the flow frequency spectrum are also considered.     

A method of examining a pair of interdependent integral equations : PMM vol. 37, n6, 1973, pp. 1078–1086

The paper deals with the method of inverting two singular integral equations of the first and second kind, respectively, possessing a definite structure. The equations as well as their solutions are obtained on the basis of analyzing a specific mixed problem of the potential theory for a quadrant.     

Flows op a reacting mixture in laval nozzles under conditions op a quasi-frozen process : PMM vol. 39, n 6, 1975, pp. 1068–1072

Flows of a chemically active gas mixture are considered in a small region of a La val nozzle, where their mode changes from subsonic to supersonic (the frozen speed of sound is considered) are analyzed. Continuous solutions and solutions with shock waves are derived. Conditions of shock-free flows are obtained.     

On the correctness of the approximate investigation of synchronous machine rotor swinging : PMM vol. 37, n6, 1973, pp. 1007–1014

We consider the complete system of equations for the dynamics of a synchronous machine with two windings on the rotor. We indicate the conditions under which the original system of equations can be reduced to the equation of motion of the rotor. The conditions for rotor selfoscillations to arise are determined as a result of investigating this equation. The complete system of equations for the dynamics of a synchronous machine containing equations describing the electrical responses and equations for the rotor's mechanical motion are obtained in [1]. Transient responses in electric circuits were investigated next, as was the expression for the electromechanical moment under a constant rotation velocity of a rotor with one circuit, e. g., field winding. However, in many of the later works the electrical equations were used only for finding the electromechanical moment under a constant spin rate of the rotor, and the problem was then reduced to the study of the equation for the rotor's mechanical motion [2, 3]. Here the conditions for which such an analysis is admissible were not mentioned. It was established that the swinging of a synchronous machine's rotor can be revealed in the form of selfoscillations. Vlasov [4] has investigated the equation of motion of a rotor and, under the assumption of a small parameter in the first derivative term, has found the conditions for the excitation of selfoscillations. Investigation in this same direction was carried out in [5]. However, in the investigation of the selfoscillations Vlasov did not examine the responses in the electrical circuits, while the expression for the electromechanical moment was obtained from power considerations. Other works have used particular expressions for the electromechanical moment, which can not explain the selfoscillation phenomenon.     

Axisymmetric contact problem for an elastic inhomogeneous half-space in the presence of cohesion : PMM vol. 37, n6, 1973, pp. 1109–1116

There is obtained the exact solution of the axisymmetric contact problem on the indentation of a circular punch into an elastic half-space having a variable modulus of elasticity E = E_vz^v (0 v < 1) in the case of the presence of complete cohesion.     

On the game theory approach to problems of optimization of elastic bodies : PMM vol. 37, n6, 1973, pp. 1098–1108

Problems of optimization of elastic bodies are considered usually in deterministic formulation, and for their solution the methods of variational calculus and the theory of optimal control are applicable (c.f., e.g., [1] and [2–4]). In the present paper there are considered those cases when either the complete information concerning the applied loads is not available,or it is known that the structure may be subjected subsequently to various loads of a certain class. The formulation is given of the problem of the determination of the shape of the elastic body, optimal for a class of loads, and there is indicated a general scheme for its solution based on the “minimax” approach used in the theory of games. Problems of optimization of elastic beams are considered and as a result of their solution certain features of optimal shapes are exhibited.     

Stability of flow of a liquid crystal layer on an inclined plane : PMM vol. 38, n 6, 1974, pp. 1031–1040

The framework of the linear mechanics of liquid crystal media [1] is used to study propagation of waves in a layer of a nematic liquid crystal (NLC) on an inclined plane, in a magnetic field, for three different cases of orientation of the anisotropy axis, namely orthogonal to the inclined plane, parallel to the inclined plane and orthogonal to the plane of flow. Such orientations of the anisotropy axis are realized in practice in the course of special machining of solid surfaces [2]. Exact solutions of the equations of motion are obtained describing the steady flow of the layer, and the behavior of small plane perturbations is studied. It is shown that two types of plane waves can propagate in a layer of the nematic mesophase, namely, the surface and the orientational waves. In the case of long surface waves the formulas for the critical Reynolds number are obtained. For the orientational waves a sufficient criterion of stability of the flow in the layer is obtained for two cases. The influence of the magnetic field and of the rheological parameters of NLC on the character of propagation of the first and second type waves is investigated.From amongst the papers dealing with wave propagation in NLC, we draw the readers' attention to [3] which deals with the longitudinal, shear and torsional waves in a liquid crystal domain and obtains the corresponding dispersion relationships.     

On the Bubnov-Galerkin method in the nonlinear theory of vibrations of viscoelastic shells : PMM vol. 37, n6, 1973, pp. 1117–1124

Results in [1] are extended to the case of vibrations of shallow and nonshallow viscoelastic (and elastic) shells. A uniqueness theorem is proved in a somewhat broader class of functions than in [1].     

Case of a generating family of quasi-periodic solutions in the theory of small parameter : PMM vol. 37, n6, 1973, pp. 990–998

We consider the problem of the existence and the stability in-the-small of periodic solutions of systems of ordinary differential equations with a small parameter μ, which in the generating approximation (μ = 0) admit of a family of quasi-periodic solutions (we are concerned only with the solutions belonging to the indicated family when μ = 0). The case to be investigated is in a specific sense a more general case of the unisolated generating solution in the small parameter theory and, therefore, includes everything previously treated by Malkin [1], Blekhman [2], and others. The main difficulty in the investigation is the presence of a multiple zero root in the characteristic determinant of the problem's generating system, to which both simple as well as quadratic elementary divisors [3] correspond. This fact predestines the presence of three groups of stability criteria for the solution being examined. The method for constructing these criteria, proposed here, assumes, in contrast to a previous one [1], the preliminary determination of not only the generating approximation but also the first one to the desired periodic solution. Particular aspects of the general “mixed” problem treated here were studied earlier in [4, 5].     

Method of successive approximations for three-dimensional laminar boundary layer problems (locally self-similar case) : PMM vol. 37, n6, 1973, pp. 974–983

We present an analytical method for the computation of problems of incompressible boundary layer theory based on an application of the method of successive approximations. The system of equations is reduced to a form suitable for integration. Parameters characterizing the external flow and the body geometry are contained only in the coefficients of the system and do not enter into the boundary conditions. The transformed momentum equations are integrated across the boundary layer from a current value to infinity with the boundary conditions taken into account. If the integration is made from zero to infinity, then the equations pass over into the Kármán relations. Integrating the system of equations a second time, using the boundary conditions at the wall, we obtain a system of nonlinear integro-differential equations. To solve this system of equations we apply the method of successive approximations. To satisfy the boundary Conditions at infinity we introduce, at each step of the iterations, unknown “governing” functions. From the conditions at the outer side of the boundary layer we obtain additional equations for their determination. With the iterational algorithm formulated in this way, the boundary conditions, both on the body and at the outer side of the boundary layer; are satisfied automatically.We consider a locally self-similar approximation. In this case, relative to the “governing” functions, we obtain an algebraic system of equations. We write out the solution in the first approximation. The results obtained in the first approximation are compared with the results of finite-difference computations for a wide range of problems. The results obtained in this paper are compared with those obtained in [1] for the flow in the neighborhood of a stagnation point. An indication is given of the nonuniqueness of the solutions of the three-dimensional boundary layer equations.