On a Malkin — Massera theorem : PMM vol. 43, no. 6, 1979 pp. 975–979

The results obtained in [1, 2] are complemented by an assertion on asymptotic stability uniform with respect to t₀, x₀, and also are extended to the problems of asymptotic stability with respect to a part of the variables and of optimal stabilization with respect to a part of the variables.     

On some hydrodynamic models of drainage : PMM vol. 43, no. 6, 1979, pp. 1046–1057

A single drain model is used for investigating with due regard to physical parameters the local depression appearing over a drain in a stratum of unbounded depth and extension under certain conditions, one of which is the increase of the drain discharge rate. Such increase is shown to be limited to some value (for given width of the section free of flooding and depth of the drain sink) that corresponds to the limit case [1] in which the depression curve has a cusp.     

On stability in one critical case : PMM vol. 43, no. 6, 1979, pp. 963–969

The problem on the stability of the trivial solution of an autonomous system of ordinary differential equations is solved in the critical case of one zero root, m pairs of pure imaginary roots, and q roots with negative real parts. It is proved that the presence of the zero root, as a rule, leads to instability, which can be detected already from the form of the second-order series expansion of the right hand sides of the equations. In the degenerate case necessary and sufficient stability conditions have been indicated for a model (simplified)system; it is shown that the absence of additional degeneracy the instability of the original system follows from that of the model. Sufficient conditions for the asymptotic stability and instability of the original system have been obtained under the fulfilment of the necessary stability conditions for the model system.     

Stratification method in the theory of thin shells : PMM vol. 43, no. 6, 1979, pp. 1058–1064

It is shown that on the basis of certain simplifications induced in the physical and geometrical dependences, such a “stratification” of a shell can be achieved for which the fibers of each of two layers will be deformed just as thin rods whose axes agree with the lines of principal curvature of the shell middle surface. The approach to analyzing shells on the basis of the relationships to be obtained below is called the “stratification method”.     

On instability in the presence of several resonances : PMM vol. 43, no. 6, 1979, pp. 970–974

The equilibrium position stability of an autonomous system of ordinary differential equations is considered in the case of n pairs of pure imaginary roots with the simultaneous presence of several resonances. It is shown using Chetaev's theorem [1] that when among the solutions of the model system there are increasing solutions of the invariant ray type, the complete system is Liapunov unstable.     

Existence of periodic solutions of the generalized burgers system : PMM vol. 43, no. 6, 1979. pp. 992–997

The boundary value problem for Burgers equations of compressible fluid is considered. Proof is given of the existence of periodic solution as the limit of solutions of initial boundary value problems in which the instant of initial data definition tends to minus infinity.     

Calculation of pressure in the linear problem of vibrator in a supersonic boundary layer : PMM vol. 43, no. 6, 1979, pp. 1014–1028

The supersonic flow over a body consisting of a triangular oscillating plate the vibrator— mounted between two flat plates is investigated. The body is assumed to be thermally insulated, and the vibrator dimensions and the oscillation frequencies to be such that the flow can be defined by equations of a boundary layer with self-induced pressure [1 — 5]. The oscillation amplitude is assumed small so that these equations can be linearized. The Fourier transform of the longitudinal coordinate is used for solution derivation. The inverse Fourier transform is obtained by numerical methods. It is shown that the perturbations of flow parameters induced by the vibrator are damped upstream and downstream in accordance with an exponential law.     

Asymptotic properties of one-component model in the theory of sorption dynamics : PMM vol. 43, no. 6, 1979, pp. 1040–1045

Asymptotic properties of integro-differential equations defining a one-component one-dimensional model of multistage dynamics of sorption is considered in the limit case when one of the kinetic stages is a limiting one.     

Determination of the equilibrium boundaries of a two-layer system convection stability : PMM vol. 43, no. 6, 1979, pp. 1008–1013

The method which permits the determination of a large number of critical Rayleigh numbers and respective critical motions is applied to the determination of convection stability boundaries of a two-layer system. The method consists essentially of reducing the problem to the algebraic problem of eigenvalues by discretizing the equations by the method of finite elements or finite differences. Only the scheme of the method based on discretization by finite elements is then presented.     

On the investigation of oscillations of quasilinear systems with almost-periodic coefficients : PMM vol.43, no. 6, 1979, pp. 980–991

A method is developed for investigating the oscillations of systems with almost-periodic coefficients, based on Kamenkov's ideas [1] on the construction of stationary solutions of systems with periodic coefficients and on the separation of motions. In contrast to [1] it is assumed that under the vanishing of a small parameter μ the system's characteristic equation has, besides n pairs of pure imaginary roots, m zero roots and h roots with negative real parts. Non-resonance and resonance cases are considered. Conditions are obtained for the existence of stationary solutions with respect to terms of first order in the small parameter. An example is presented.     

On the problem of shock wave formation within a local supersonic zone : PMM vol. 43. no. 6, 1979, pp. 1117–1118

A particular exact solution of the transonic equation is considered as an extension of Frankl's solution [1] for the flow at some distance from the profile. The derived generalization defines the flow over some half-body of a gas stream at a velocity that is subsonic at infinity.     

On the carnot theorem in the theory of impulsive motion of mechanical systems : PMM vol. 43, no. 6, 1979, pp. 1114–1116

The kinetic energy variation of a system subjected to impact under conditions of the generalized Carnot theorem [1] is investigated. Proof of the theorem is given in the case of acatastatic links. It is shown that the second part of that theorem extends to systems whose interaction with the links is defined by properties which include the known conditions of ideality of the latter and, also, of zero virtual work of link reaction impulses during the regeneration time.     

Method of homogeneous solutions in the mixed problem for a finite circular cylinder : PMM vol. 43, no. 6, 1979, pp. 1073–1081

The mixed axisymmetric problem of elasticity theory on the torsion of a finite circular cylinder by a stamp is considered. The stamp is fixed rigidly to one plane face of the cylinder, the other plane face is fixed, and conditions for no displacements or stresses are given on the cylinder surface. The problem is investigated by the method of homogeneous solutions [1], which permits obtaining its approximate solution for practically any values of the parameters. Such efficiency of the method is determined by the fact that the solution of the problem reduces to investigating an infinite algebraic system of the Poincaré — Koch normal systems type. When the ratio of the cylinder height to the radius of the stamp is sufficiently large, the system coefficients, the contact stresses, and the other characteristics of the problem are evaluated to any degree of accuracy, and effective asymptotic expressions are obtained for small values of this ratio. Results of numerical computations are presented.

A solution of the problem for the case of a large value of the ratio (R − a) /h and small values of the ratio λ = h / a is obtained in [2].     

On the stability of elastic bodies with random in homogeneities under finite deformations : PMM vol. 43, no. 6, 1979, pp. 1125–1129

Stability of stochastically inhomogeneous, compressible elastic bodies with respect to small, as well as to finite perturbations, is studied in three-dimensional formulation. The bodies are under deterministic external loads and experience finite subcritical deformations.

The stability of elastic bodies with random inhomogeneilies was studied for the case of small, subcritical deformations in [1]. The basic relations for a stochastically inhomogeneous compressible hyperelastic body can be obtained from the relations for compressible hyperelastic media given in [2].     

On certain cases of determination of trajectories of plane conservative motion of a particle : PMM vol. 43, no. 6, 1979 pp. 1111–1113

A method is developed for obtaining an incomplete integral of the two-dimensional Hamilton— Jacobi equation for a particle when a particular solution that satisfies a special equation in partial derivatives is known. The equation for the trajectory of dipole particles moving with zero total energy in an arbitrary two-dimensional electrostatic field is obtain, as an example.     

Electrohydrodynamic flow in a retarding electric field with allowance for charged particle inertia : PMM vol. 43, no. 6, 1979, pp. 1029–1039

The one-dimensional electrohydrodynamic flow is considered in a retarding electric field when, owing to the inertia of charged particles, it is necessary to use the complete equation of momenta for the charged component. It is shown that in spite of the negligibly small relative volume occupies by particles in the initial cross section of the stream, there is a section in which particles move at low but finite velocity with a finite relative volume of the charged component, when the particles are retarded by an external homogeneous electric field (without taking into account the induced field). Interaction of the charged and neutral components, which may be absent in the initial cross section, is always substantial. The pressure drop required for such flow is determined. The flow is investigated with allowance for induced electric fields that diminish the effect of concentration increase of charged particles.     

Influence of geometrical nonlinearity on the waves propagating through a thin, free plate : PMM vol. 43, no. 6, 1979, pp. 1089–1094

Exact wave solutions of the equations of motion of a thin plate are obtained for the one-dimensional case of a first approximation model. The dependence of the velocity of propagation of flexural-longitudinal waves on their frequency is computed for different values of the amplitude of the bending components. The nonlinear character of the relation connecting the deformations and displacements in the theory of thin shells may give rise to effects which cannot be described in terms of the linear approximation even in those cases when the Hooke's law still holds [1]. The estimation of the magnitude of the displacements at which the effects caused by the geometrical nonlinearity become apparent, is of interest.     

On integration of the unsteady equation of heat exchange between bodies moving in fluid : PMM vol. 43, no. 6, 1979, pp. 1119–1122

The unsteady problem of convective heat exchange of bodies of a arbitrary shape moving in a perfect fluid or of drops moving in a viscous incompressible liquid, is considered in the approximation of thermal boundary layer.     

On the stability of space-periodic convective flows in a vertical layer with sinuous boundaries : PMM vol. 43, no. 6, 1979, pp. 998–1007

The problem of convection in a vertical layer with harmonically distorted boundaries is examined by perturbation theory methods for a small amplitude of sinuosity. The solutions obtained are applicable both in the stability region as well as in the supercritical region of the plane-parallel flow. The stability of the solutions found is investigated with respect to a certain class of space-bounded perturbations that are not necessarily space-periodic. The method of amplitude functions [1], generalized to the case of curved boundaries, is used. The Grashof critical number is found as a function of the period of sinuosity and the form of the neutral curve for the space-periodic motions and their stability region are obtained. It is established that if the deformation period of the boundaries is close to the wavelength of the critical perturbation for the plane-parallel flow or is twice as great, then as the Grashof number grows stability loss does not occur and the motion's amplitude changes continuously (cf. [2 — 4]). A comparison is made with the results of the numerical calculation in [5], An attempt was made in [6] to construct a stationary periodic motion in a layer with weakly-deformed boundaries, in the form of series in powers of a small sinuosity amplitude. However, the solution obtained diverges in a neighborhood of the neutral curve of the plane-parallel flow and approximates unstable motion in the supercritical region of the unperturbed problem. Flows under a finite sinuosity amplitude are calculated by the net method in [5] wherein the stability of the flows was investigated as well, but only with respect to perturbations with wave numbers that are multiples of 2π/l, where l is the length of the calculated region.     

Contact problem for two microinhomogeneous half-spaces : Pmm vol. 43, no. 6. 1979. pp. 1082–1088

The interaction between two microinhomogeneous half-spaces in a macrohomogeneous stress-strain state is considered. The perturbation method is used to solve the problem. Finite expressions for the correlation functions and dispersion of deformations are obtained in the first order approximation. Numerical values of the coefficients characterizing the stress concentration at the boundary of interaction between the half-spaces are derived. The dependence of these coefficients on the ratio of elastic moduli of the media filling the half-spaces is studied. A particular solution of this problem for the case when the elastic moduli depend on two coordinates only, was given in [1].