An example of a linear completely contrallable time-invariant system is considered to study the change of its attainability region after the transition to a reduced system with a decreased order. The original system is a triple integrator. After reduction, a new system (a double integrator) whose control belongs to the set of piecewise continuous bounded functions is considered. The attainability set

The quasistatic problem of determining the stress-strain state in a cylindrical axisymmetric layer is solved. The layer is subjected to the action of tangential forces directed along the axis of the cylinder and uniformly distributed over the outer surface of the layer whose inner surface is fixed. A hypoelasticity model for an incompressible material under finite strain is adopted. This model is based

The motion of Chaplygin’s skate on a horizontal plane with dry friction is considered. It is assumed that the friction force depends on an angle between the sliding velocity and the plane of the skate. It is shown that the time of the skate sliding can be found from a transcendental equation and depends on the initial values of the angular velocity, the sliding velocity, and the angle between the sliding

The problem of sequential loading-unloading a wheel with a deformable rod protector (tire tread) is considered. The presence of a slip zone in the contact zone with the road surface leads to different restoring forces during the loading-unloading process and to the energy losses due to dry friction. Two different approaches to finding the total normal reaction acting on the wheel from the road surface

A system of ideal polytropic gas equations written in the Lagrangian coordinates is considered on a uniformly rotating plane. For this system the first integrals corresponding to motions with uniform deformation are found. It is shown that, if the adiabatic index is equal to two, the original system consisting of four second-order nonlinear ordinary differential equations can be reduced to a single

The equations of motion of the Chaplygin sleigh on a horizontal plane with friction are obtained. A geometric interpretation of the continuous motion is given. It is proved that the motion of the sleigh stops in a finite time. Some properties of the motion process are discussed.

An exact analytical solution is obtained for the extension problem in the case of an infinitely long continuous round cylinder attached to an external hollow round cylinder. This mechanical system is considered as a composite rod whose components are made of elastic materials with different mechanical characteristics. The method of solving this problem is based on the approach proposed by the author

The problem of the explosion of a cord charge on the surface of the ground is studied within the solid-liquid formulation of the pulse-hydrodynamic model. The rectilinear segments of the soil free surface adjacent to the cord charge are inclined to the plane of the charge at certain angles. The exact solution to the problem is given. A parametric analysis is performed. The calculated profiles of the

A method is proposed and the simple analytical solutions are obtained for the diffraction problems of acoustic waves on a half-plane with the impedance boundary condition on its one side and the Neumann or Dirichlet condition on the opposite side. It is shown that the properties of these solutions are essential for the optimal design of sound barriers with sound-absorbing walls.

Some numerical results obtained by modeling the thermogas displacement of oil from a porous reservoir are discussed. The displacement is performed by a heated water-gas mixture. The injected two-component gas consists of nitrogen and oxygen. Heat, carbon dioxide, and water vapors are released during the reaction process. The heat release decreases the oil viscosity and accelerates the displacement

The minimax problem of stabilization is solved for a line of sight near a programmed trajectory. The motion of this line is described by a system of fourth-order linear differential equations. In this problem, the perturbations are considered as the deviations of the initial position from zero and as time-varying perturbations. The stabilization is performed by a linear feedback. The feedback coefficients

It is assumed that a certain reference frame is inertial for a system of moving and interacting bodies called a large system. In the framework of classical continuum mechanics, some necessary and sufficient conditions are obtained for the existence of a reference frame for a subsystem of this large system considered as an independent large system. The motion of such a new reference frame with respect

The integrability of certain classes of dynamical systems on the tangent bundle of a multidimensional manifold is shown. In this case, the force fields with variable dissipation generalize those considered earlier.

The dynamic behavior of Bernoulli-Euler and Timoshenko beams subjected to a moving concentrated load is considered. Deflection expressions are found in the form of convergent series for a pretensioned beam and for a beam on an elastic foundation. We determine the parameter ranges for which the results of these two models are coincident. It is shown that, under certain conditions, a resonance is possible

The splitting of initial boundary value problems in the theories of elasticity is considered for some anisotropic media. In particular, the initial boundary value problems of the micropolar classical theory of elasticity are represented using the tensor-block matrix operators (or tensor operators). In the case of isotropic micropolar elastic media known also as isotropic or transversally isotropic

The nonstationary processes are considered in a prestressed medium for the case of a sudden viscoelastic break in continuity under the conditions of longitudinal shear. It is assumed that the break proceeds along a semi-infinite strip coincident with the plane of maximum shear stresses. The problem is solved analytically in terms of displacements by the Wiener-Hopf method.

The motion of a puck on a horizontal plane rotating about a vertical axis with dry friction is considered. It is assumed that the Coulomb law of dry friction is locally valid at each point belonging to the lower surface of the puck. The resultant force and friction torque are determined according to the dynamically consistent model of contact stresses. This problem generalizes the problem of motion

The possibility of constructing a mathematical model of disperse systems is discussed. This model is similar to those used in the theory of ideal gases.

The accuracy of stabilization algorithms for a linear system is estimated using maximin testing methods on a finite time interval in the presence of initial and time-varying perturbations.

An algorithm is developed to numerically study the flexible rods (hoses) loaded by concentrated forces and situated in an air or liquid flow. The developed algorithm allows one to solve the static problems with consideration of a liquid flow in a hose. This algorithm can be used to determine an equilibrium rod shape and an axial force in the rod when the velocity vector of the external liquid flow

The equivalence between a possible formulation of an elasticity theory problem in terms of stresses and its classical formulation is shown for rather general domains.

An equation describing a family of stress relaxation curves with arbitrary strain programs at the initial stage generated by the Rabotnov nonlinear constitutive relation with two arbitrary material functions is derived and studied analytically. The effect of the initial stage duration and shape on the properties of relaxation curves is analyzed. Their deviations from each other and from the relaxation

The mutual influence of three-dimensional disk-shaped cracks located in parallel planes of an elastic medium is studied. The medium is under tensile stress directed perpendicular to the crack planes. The cracks are modeled by mathematical cuts of the continuous medium with the possibility of a strong discontinuity of the displacement field on the faces of the cut. The discontinuous displacement method

The programmed motion of an aircraft is considered. The optimization stabilization parameters are sought for this motion. The minimax stabilization problem is formulated and solved. The best parameter is found on the chosen grid and the necessary optimality condition is verified.

The unstable equilibrium positions of four mechanical systems are considered. The following systems are chosen: the system with one degree of freedom (an inverted mathematical pendulum), the systems with two degrees of freedom (an inverted spherical pendulum), the system with a countable number of degrees of freedom (an elastic beam loaded by a compressive force greater than the Eulerian critical load)

The theoretical and experimental results are compared in the classical problem on regular two-dimensional waves in heavy viscous liquid films down a vertical plane. Some similarity transformations are used to perform a comparative analysis of numerical methods to study the nonlinear waves formed in a film during the spatial and time development of disturbances in the main steady flow.

The spouting of vertical submerged axisymmetric jets from under the free surface in relatively narrow channels is experimentally and numerically studied. The existence of self-oscillation regimes with transverse displacements of the free-surface elevation is found for wide ranges of constitutive parameters. Dependencies of the self-oscillation period on the velocities of jets and the initial value

In this paper it is shown that the semi-major axes of the orbits of neighboring planets and of the orbits of large satellites of some planets in the Solar system are close to the orbit radii in the optimal solutions of the problem on a single-impulse transfer in a planetary system from a circular orbit to infinity with a gravitational maneuver.

The problem of motion of a gyroscope in a gimbal suspension is discussed. All steady motions of the system, their stability and branching conditions are given. The results are presented in the form of an atlas of bifurcation diagrams.

The isentropic motion of a perfect ideal gas on a constant homogeneous supersonic background is considered. A two-dimensional stationary potential problem of gas dynamics is solved to study the interaction of three traveling waves whose amplitudes and phases are slowly varying in the direction of the background flow in the case when the sum of “harmonic” phases is exactly equal to zero. The equations

A brief historical review of creating the ecological wave propulsors mounted on ships is presented. A particular attention is given to the arrangement of such devices. Experimental data on model tests concerning the positioning of wave propulsors on a ship to increase their efficiency are discussed.

A number of results obtained during the numerical finite element analysis of a procedure to test the ring specimens of polymer materials under internal pressure are discussed. The internal pressure is produced by the compression of an inset made of an incompressible material in the cavity of the specimen under test. The analysis is performed for the ring polyarylate specimens. The stress distributions

Reducing the aerodynamic drag of bodies of revolution at supersonic speeds due to the creation of a bottom thrust is an important subject of aeromechanics. In the framework of this problem, it is proposed to perform numerical experiments on a model with a detachable bottom. It is assumed that the detachable bottom is affected by the main air flow jets separating from the surface of the body of revolution

The process of ejecting a liquid from a finite-size vessel with an inclined wall by a submerged wall jet is considered. The character of flows arising in the vessel and the intensity of liquid ejection are numerically studied for various values of jet thickness and velocity. The numerical method in use is verified by a number of experiments. It is found that, at the last stages of ejection, in the

A phenomenological model of brittle fatigue fracture in metals and alloys is discussed in the case of proportional (simple) loading. The model is formulated in the framework of the physico-mechanical approach as a system of hypotheses concerning the probability of defect evolution at various scale-structural levels, such as micro- and macrocracks. The described constitutive relations are based on this

The application of galvanic vestibular stimulation is considered for the galvanic correction of pilot’s vestibular activity during visual flight control on a flight-dynamic stand and in extreme flight situations.

Some approaches to the axiomatic formulation of theoretical fundamentals in continuum mechanics are considered. The basic concepts, laws, and hypotheses in the classical theory of continuum mechanics and their modifications for its nonclassical versions are discussed. In the framework of the classical version of the rational theory, a number of axioms are proposed for the general theory of constitutive

The effect of various weaving types for fabric composites on the low-speed (up to 350 m/s) penetration into multilayer fabric barriers made of aramid fibers is studied. The geometric properties of plain and twill weaves are taken into account on the basis of experimental data showing that the warp and weft threads may have different elastic and ultimate properties. Significant differences in the parameters

The calibration of a strapdown inertial navigation system is considered in the case of varying temperature. A parameterized model of measurement errors is used to calibrate the inertial sensors. In addition to usual standard parameters, this model contains the coefficients in the temperature dependence. In the author’s previous publications, it is shown that it is possible to estimate these coefficients

The vector-matrix Shulgin’s equations are used to stabilize the steady motions of mechanical systems with nonlinear geometric constraints in the case of incomplete information on the state. The momenta are introduced only for the cyclic coordinates that are not used to control. Three variants of the measurement vector are used to prove a theorem on the stabilization of control with the help of a part

The filtration of a supercritical brine flow during the magma chamber degassing is numerically simulated with consideration of quartz transport and deposition. It is shown that the quartz deposition and the accompanying decrease of permeability reduce the size of the concentrated brine lens formed above the magma chamber. The effect of rock hydraulic fracture on the lens formation is studied. An unsteady

The explosion of a wedge-shaped cord charge on the ground surface is considered in the solid–liquid formulation of the pulsed-hydrodynamic model. The general solution to this problem is obtained. A parametric analysis is performed. The dependence of the shape and size of the explosion crater on the constitutive parameters is analyzed.

The motion of a single melting spherical particle in its own melt is considered. A formula for the thrust appearing due to flow around the melting particle is derived. The self-similar melting problem is solved for a particle at rest in a quiescent melt to determine the radius of the particle. For a small Reynolds number, it is shown that there exists a critical temperature difference between the particle

The dynamics of a mobile vehicle with three omni wheels is considered for the case when the vehicle moves on a fixed perfectly rough plane by inertia. In order to study the effect of wheel roller inertia on the stability of the rectilinear motion of the vehicle, the fore wheel is modeled by the following rigid bodies: a wheel disk and a supporting roller such that the roller does not slip with respect

A particular case of the game problem formulated in our previous paper is considered. As a model of a moving platform, we use an extended model of the Dubins car. An existence condition for a saddle point and a method of its finding are proposed. A method of determining the strategies and trajectories of players is discussed.

The physical laws characterizing the relation between stresses and strains are considered and analyzed in the general modern theory of elastoplastic deformations and in its postulates of macroscopic definability and isotropy for initially isotropic continuous media. The fundamentals of this theory in continuum mechanics were developed by A.A. Il’yushin in the mid-twentieth century. His theory of small

A generalized theory of stress and strain tensor measures in the classical continuum mechanics is discussed: the main axioms of the theory are proposed, the general formulas for new tensor measures are derived, arid an energy conjugate theorem is formulated to distinguish the complete Lagrangian class of measures. As a subclass, a simple Lagrangian class of energy conjugate measures of stresses and

Two solutions of three-dimensional Navier–Stokes equations are studied numerically. These solutions describe the fluid motion in a plane channel, are of the traveling-wave form, and are periodic in the streamwise and spanwise directions. It is shown that, in each solution, the oscillations arise as a result of linear instability in the streamwise averaged velocity field. This instability is due to

The problem of description and classification of rapid eye movements called saccades is considered. It is well known that the saccades are of various types. A classification of saccades is proposed according to the presence and types of pre-saccadic and post-saccadic movements. An algorithm is developed to determine the parameters that can be used to formally assign the saccade to one of the given

The static loading of a wheel pair with a deformable periphery is considered when the wheels are mounted on a common axle with a non-zero camber angle. A tire tread (protector) is modeled by a set of elastic rods interacting with the motion plane according to the dry friction law. The wheel camber effect on the slip in the contact zone and on the magnitude of reaction forces from the road is studied

We consider a natural Lagrangian system on which an additional holonomic rheonomic constraint is imposed; the time dependence is included in this constraint by a parameter performing rapid periodic oscillations. Such a constraint is said to be a vibrating constraint. The equations of motion are obtained for a system with a vibrating constraint in the form of Hamilton’s equations. It is shown that the

A simple method is proposed for solving the problems of diffraction of plane sound waves on a half-plane with different-type boundary conditions on its surfaces (the Neumann condition on one surface and the Dirichlet condition on the opposite surface). In contrast to existing techniques, the proposed method allows one to obtain analytical solutions valid near the half-plane edge and at far distances

The gravity-forced motion of an ideal incompressible fluid of infinite depth is studied when a periodic pressure is applied to the surface of the fluid. This problem is solved on the basis of the small amplitude wave theory. The analytical solutions for the velocity potential, the velocity field, and the shape of the free surface are found. An expression for the horizontal force is obtained in the

The equations of motion for a dynamically symmetric n-dimensional fixed rigid body-pendulum situated in a rioricoriservative force field are studied. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of an incident medium. The complete list of transcendental first integrals expressed in terms of finite combinations of elementary functions

In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of volume and surface forces of general form. À notion of internal force factors is introduced. The equations for force factors (the equilibrium equations in the median

A procedure of reducing the three-dimensional problem of elasticity theory for a rectilinear beam made of an anisotropic iuhomogeueous material to a one-dimensional problem on the beam axis is studied. The beam is in equilibrium under the action of volume and surface forces. The internal force equations are derived on the basis of equilibrium conditions for the beam from its end to any cross section

A mathematical formulation of the line-of-sight control problem is proposed for the case when this line is directed at a target. An operator situated on a moving platform controls the line of sight using the data received from video images. Some functionals determining the quality of control by the operator are introduced. It is proved that, in the case of plane motion of the platform and an infinitely

A model describing the shape of the stationary segment of a heavy flexible inextensible thread moving in a fixed vertical plane down to a given depth from a fixed position to a fixed position is considered. The parametric equations of the stationary curve are derived. The shape of the stationary segment and its properties are in a qualitative agreement with those observed in experiments.

Some boundary conditions used in the modern theory of disperse systems and the associated questions concerning the effective viscosity and the effective thermal conductivity are discussed.

A theory of constitutive relations describing the resistance of bodies to deformation is developed. This theory takes into account the presence of internal body forces and internal kinematic constraints. A number of axioms and a general reduced form of constitutive relations for classical media are proposed. For simple bodies it is proved that the Il'yushin and Noll constitutive relations are equivalent