On a particular form of stabilized capillary-gravitational waves of finite amplitude at the surface of fluid over an undulating bed : PMM vol. 37, n6, 1973, pp. 1020–1034

The problem of stabilized plane capillary-gravitational waves of finite amplitude at the surface of a stream of perfect incompressible fluid flowing over an undulating bed and subjected to pressure periodically distributed along the surface and defined by some infinite trigonometric series is considered. The intersection of the bed with a vertical plane is assumed to be a periodic curve, called the bed line, defined by some infinite trigonometric series. The problem is rigorously formulated and reduced to the solution of a system of nonlinear integral and transcendental equations. The solution is constructed in the form of series in powers of a small dimensionless parameter to which amplitudes of the first harmonics of the bed line and of the surface pressure wave are proportional. An approximate equation is derived for the wave profile.The particular case is considered, when the length of the bed line wave arc is equal to the length of the stabilized free wave line corresponding to the specified flow velocity over a horizontal flat bed and constant pressure along the surface. In such case the parameter of the integral equation is equal to one of the eigenvalues of the kernel of that equation and the solution is constructed in the form of series in powers of the cube root of the small parameter mentioned above.A similar problem but for constant pressure along the surface was considered by the author in [1, 2] and in his paper presented at the 13-th International Congress on Theoretical and Applied Mechanics (Moscow, 1972 [3]).Another similar problem of capillary-gravitational waves over an undulating bed was considered in [4], where besides the topological proof of the existence and uniqueness of solution the algorithm for constructing the latter is given, but the calculation of approximations is only outlined and the mechanical meaning of solution is not investigated in depth.Unlike in [4] the equation of the bed line and the expression for pressure at the surface are specified here in a form which makes it possible to express any approximations in the form of finite sums, and an analysis of the fundamental system of nonlinear integral and transcendental equations by the LiapunovSchmidt analytical methods and their developments is presented.     

On stability of equilibrium of nonholonomic systems : PMM vol. 39, n 6, 1975, pp. 1135–1140

The problem of stability of motion of nonholonomic systems was first considered by Whittaker in [1], and developed in [2–7] et al. The most general results in investigating the stability of equilibrium of conservative nonholonomic systems and in clarifying the influence of the dissipative forces on this stability, were obtained in [5]. In the present paper we give a further generalization of the results obtained in [5].     

The nonlinear problem of unsteady filtration of heavy fluid with a free surface : PMM vol. 39, n 6, 1975, pp. 1017–1022

A solution is derived for the problem of unsteady motion of heavy fluid with a free surface in the vertical plane in a porous medium. Such problems are encountered in irrigation and land improvement schemes in connection with the filtration of ground waters. To use numerical and approximate methods for obtaining a solution of this fairly difficult problem one must be sure of its existence. The case when the heavy fluid occupies, at the initial instant of time, a finite region, is considered. An earlier investigation of this problem by the author [1] was based on some other assumptions with the heavy fluid occupying a semi-infinite region.     

On stability of autonomous systems with internal resonance : PMM vol. 39, n 6, 1975, pp. 974–984

When investigating the stability of the trivial solution of an autonomous system of ordinary differential equations in the critical case of n pairs of pure imaginary roots an essential role can be played by the presence of integral linear dependences between the system's frequencies or, in other words, by the internal resonance. Various special cases of this problem were examined in [1–6]. Our aims are: to obtain a special (normal) form of the differential equation system with internal resonance of most general form in it; to ascertain the conditions under which the presence of internal resonance does not permit the application stability investigation methods developed for resonance-free systems; to solve the stability problem in one of the most important cases of odd-order internal resonance, generalizing the preceding investigations. In the solution of the last problem the necessary and sufficient conditions are given for the stability of the model (simplified) system. Using Chetaev's theorem we show that as a rule the instability of the original system follows from the Instability of the model system. Cases of structurally-unstable instability (^*) for which the model system does not resolve the problem of stability are outlined. The results obtained are extended, in particular, to Hamiltonian systems.     

On the determination op forces op constraint reaction : PMM vol. 39, n 6, 1975, pp. 1129–1134

The equations of motion of mechanical systems with multipliers are reduced to the form enabling the separation of these equations into two groups, the first group describing the motions of the system, and the second group defining the multipliers. Each multiplier is determined independently of the remaining multipliers, and this makes it easy to assess the dynamic effect of each constraint on the system. On the basis of this approach, we study the following problems: determination of the constraint reactions [1], study of the motion of controlled systems with prescribed constraints [2, 3] and utilization of the method of nonholonomic mechanical systems in the case when the first integrals exist [4].     

On integrals of equations of unstable nearly self-similar flows : PMM vol. 39, n 6, 1975, pp. 1060–1067

One-dimensional or nearly one-dimensional unstable motions of perfect gas are considered. Integrals admitted by the system of equations defining such motions are examined. Since the existence of integrals is associated with some law of conservation, i. e. with some divergent form of presentation of equations of the input system, it is possible by examining all divergent equations of gasdynamics to derive certain new integrals not previously considered.     

Time-optimal control synthesis for a fourth-order nonlinear system : PMM vol. 39, n 6, 1975, pp. 985–994

On the basis of Pontriagin's maximum principle we establish the structure of the optimal control and of the optimal trajectories, using the properties of the system being analyzed. We propose a rule for the construction of the program control satisfying the maximum principle. In the case when the terminal state lies outside some bounded region we prove that the rule mentioned determines the optimal control and permits us to solve the synthesis problem.     

Waves in an inhomogeneous fluid in the presence of a dock : PMM vol. 39, n 6, 1975, pp. 1140–1142

We investigate the propagation of waves generated by oscillations of a section of the bottom of a tank through a two-layer fluid, in the presence of a dock. Wave motions in an inhomogeneous fluid generated by displacement of a section of the bottom of a tank were studied in [1] where the upper surface of the fluid was assumed either to be completely free, or completely covered with ice. In the present paper we use the method given in [2] to investigate a similar problem under the assumption that the fluid surface is partly covered with an immovable rigid plate. The expressions obtained for the velocity potential are used to determine the form of the free surface and of the interface. We show that when the fluid is inhomogeneous, the wave amplitude on the free surface increases, while the presence of a plate reduces the amplitude of the surface waves, as well as of the internal waves in the region between the plate and the oscillating section of the bottom.     

Singular perturbations in one-dimensional unsteady motion of a real gas : PMM vol. 39, n 6, 1975, pp. 1051–1059

Navier-Stokes equations for one-dimensional motion of gas are reduced to a special dimensionless form convenient for investigations involving a perturbation front. In new variables the transition from limit conditions of motion of an inviscid non-heat-conducting gas to the case of small but finite coefficients of viscosity and thermal conductivity, which is simulated by a perfect gas with singular perturbations induced by the indicated dissipative factors. We establish the inevitability of existence of two regions of singular perturbations, the neighborhood of the perturbation front and that of the point (line, surface) where the investigated motion is generated. The derivation of equations for both boundary layers, which is valid for a fairly general statement of problems of this kind, is presented and conditions of merging with the external (adiabatic) flow are formulated. Examples of computation of motion in boundary layers in problems of piston and point explosion are presented.     

On the stability of the natural unstressed state op viscoelastic bodies : PMM vol. 39, n 6, 1975, pp. 1110–1117

Within the framework of the Cauchy problem, a class of models of a linearviscoelastic body subjected to the stability principle of the natural unstressed state state of viscoelastic bodies (Principle Y) is isolated in [1], The principle Y is formulated as follows. Let the boundary conditions be such that the appropriate elasticity theory problem has a zero solution. If a viscoelastic body is free of external loads at each instant t > 0, then for every initial state, strain of the body vanishes as t → ∞. The principle Y is called partial if it is satisfied only for some particular class of viscoelasticity problems. Sufficient conditions for compliance with the partial Y principle are obtained in this paper for models of viscoelastic bodies within the framework of the fundamental initial-boundary value problems for finite bodies.     

The legendre condition in optimum problems of supersonic gasdynamics : PMM vol. 39, n 6, 1975, pp. 1032–1042

The necessary Legendre condition for problems of optimum (in the sense of minimum wave drag) supersonic flow past bodies is obtained. Plane and axisymmetric flows are considered on the assumption of imposition of isoperimetric constraints of a general form. Shock-free flows and flows with attached shock waves are investigated. The method here proposed is used for deriving the second order condition in the particular case when it is possible to pass to the reference contour, and which has been earlier obtained by Shmyglevskii [1] and then by Guderley and others [2].     

On the influence of gyroscopic forces on the stability of steady-state motion : PMM vol. 39, n 6, 1975, pp. 963–973

The question of the influence of gyroscopic forces on the stability of steady-state motion of a holonomic mechanical system when the forces depend upon the velocities of only the position coordinates was answered by the Kelvin-Chetaev theorems [1] on the influence of gyroscopic and dissipative forces on the stability of equilibrium. However, if the gyroscopic forces depend as well on the velocities of the ignorable coordinates, then their influence on the stability of steady-state motions can, as the two problems in [2] show, prove to be entirely different from the influence of gyroscopic forces depending only on the velocities of the position coordinates. In this paper we investigate the influence of gyroscopic forces depending linearly on the velocities of the generalized coordinates, including the ignorable ones, on the stability of the steady-state motion of a holonomic conservative system. We prove that when the gyroscopic forces applied with respect to the ignorable coordinates are given as total time derivatives of certain functions of the position coordinates, the gyroscopic forces can both stabilize as well as destabilize the steady-state motion. Under certain conditions, this influence is also preserved for the action of dissipative forces depending on the velocities of only the position coordinates. In the case of action of dissipative forces depending also on the velocities of the ignorable coordinates, we have indicated the stability and instability conditions of the steady-state motion. Examples are considered. In conclusion, we discuss the conditions under which the application of gyroscopic forces to the system is equivalent to adding terms depending linearly on the generalized velocities to the Lagrange function.     

On the behavior of solutions of equations for double waves in the neighborhood of the quiescent region : PMM vol. 39, n 6, 1975, pp. 1043–1050

The structure of solutions of gasdynamic equations is investigated in the case of unsteady double waves in the neighborhood of the quiescent region. A general concept of double waves is presented in the form of special series with logarithmic terms. Results of numerical computations are given.The problem of determining the flow of plane and three-dimensional waves separated from the quiescent region by a weak discontinuity was considered in [1–3], where approximate solutions were derived for that neighborhood, and the formulation of boundary value problems required for solving the equation for the analog of the velocity potential in the hodograph plane was investigated.The more general problem (without the assumption of the degeneration of motion) of arbitrary potential flows of polytropic gas adjacent to the quiescent region and separated by a weak discontinuity was considerd in [4–8]. Solution of that problem was obtained in the form of special series in powers of the mo dulus of the velocity vector r in the space of the time hodograph. The value r = 0 corresponds to the surface of weak discontinuity that separates the perturbed motion region from that at rest. Some applications of derived solutions to problems such as the motion of a convex piston and the propagation of weak shock waves were also investigated in those papers. Convergence in the small of obtained series was proved in [9]. However the attempts of constructing series in powers of r, which were used in [4–8] for the presentation of equations of double waves in the neighborhood of the quiescent region, proved to be unsuccessful.Although parts of expansions in series in powers of r (accurate to within 0 (r²)), were constructed in [1–3], it was found that the coefficient at r⁸ in equations for double waves cannot be determined owing to the insolvability of its equation. This is related to the fact that the surface r = 0in the case of equations for double waves is simultaneously a line of parabolic degeneration and a characteristic.The object of the present note is the formulation of solutions of equations for plane unsteady double waves in the neighborhood of the quiescent region. Parts of the derived series, which generally are nonanalytic functions of r, can be used for defining flows at small r in particular those downstream of two-dimensional normal detonation waves [10] or in problems of angular pistons [11]. The method used for the derivation of series can be also applied in investigations of threedimensional self-similar flows with variables x₁/x₃ and x₂/x₃ (steady flows) or x₁/t, x₂/t and x₃/t (unsteady flows). However it was not possible to obtain in such cases regular series in powers of r.     

State of stress in a plat circular ring with a crack : PMM vol. 39, n 6, 1975, pp. 1118–1128

The stress distribution in a circular isotropie ring with a crack on part of the concentric circle is investigated. A system of functional equations governing the coefficients of the complex Fourier series expansion of the stresses acting on the circle on which the crack is located is obtained. The solution of the mentioned system of equations is obtained by using a factorization method, which permitted reduction of the initial system of equations to two coupled infinite systems of algebraic equations. The possibility of using the method of truncation to solve these systems is proved. The singularity originating in the neighborhood of ends of the crack in the formulas governing the stresses is isolated. Stress intensity coefficients for the effect of a uniform load on the external contour are presented.     

Small-parameter method for constructing approximate strategies in a class of differential games : PMM vol. 39, n 6, 1975, pp. 1006–1016

We examine a class of problems in which the pay-off is some function of the terminal state of a conflict-controlled system. When the opportunities of one of the players are small in relation with the opportunities of the other, we propose methods for constructing approximate optimal strategies of the players, based on solving the Bellman equation containing a small parameter. We have shown that the players' approximate optimal strategies can be constructed if the solutions of the corresponding optimal control problems are known. The error bounds for the methods are proved and examples are considered. The arguments used rely on the results in [1–6] on the theory of differential games and on [7–11] devoted to optimal control synthesis methods for systems subject to random perturbations of small intensity.     

Optimal control of the motion of a quasilinear oscillatory system by small forces : PMM vol. 39, n 6, 1975, pp. 995–1005

We solve certain optimal control problems for the motion of a single-frequency oscillatory system which in the unperturbed state consists of an arbitrary number of oscillating elements. The solution is performed in the first approximation with respect to a small parameter . We assume that the frequency depends upon slow time, while the control goes only into the perturbing terms, so that the system is formally weakly controllable [1], But since the time interval over which the process evolves is a quantity ˜1/, all the controlled quantities are able to vary substantially [2, 3], i.e. we investigate the case, interesting in practice, of small but protracted control forces. As mechanical examples we calculate some optimal control problems for the oscillations of systems of the plane oscillator type, etc.