Abstract The paper deals with the one-parameter family of Gordon–Schowalter objective derivatives including the Oldroyd, Cotter–Rivlin, and Jaumann derivatives. For a simple shear, movable bases are found in which the considered differential operators are reduced to the total time derivatives of the tensor components. For all derivatives of the family under consideration, except for Oldroyd and Cotter–Rivlin

Abstract Based on the advanced convergence method developed by the authors, torsional oscillations of a circular disc that is fixed on a shaft are investigated numerically and analytically. The thickness of the disc depends on the radius. The internal boundary of the disc is fixed on a shaft while the external one is load-free. The first few eigenfrequencies of torsional oscillations are obtained for

Abstract An approximate analytical solution to the equations of the physical theory of meteors is obtained for a meteoroid moving as a single body under the assumption of constant ablation parameter. The approach allows predicting the velocity and mass of the meteoroid at any point of its trajectory. The accuracy of the solution is estimated. A numerical-analytical method for solving the equations

Abstract In this paper the displacement discontinuity method of high-order accuracy and its application to the problems of fracture mechanics are considered. In common practice of applications of boundary element methods, the methods with the piecewise-constant function of boundary displacement are often used. Their advantage against other algorithms is the simplicity of calculation scheme with a rather

Abstract The problem of motion of three connected rigid bodies in a homogeneous field of gravity forces (the generalization of the problem about motion of a gyroscope in gimbal suspension) is discussed. All steady motions of the system, their stability conditions, and branching are found. The results are presented in the form of bifurcation diagrams.

Abstract A linear mathematical model of forced oscillations of a gas bubble in viscous Newtonian fluid is developed in this paper. The natural frequency of oscillations of the gas bubble is found. A numerical analysis of the resonant effects of gas bubble oscillations within the media is carried out and a comparison of the obtained results with the existing classical ones for non-viscous media is discussed

Abstract The experimental results of the processes of complex loading along helical strain trajectories are used to find out that the response to the helical strain trajectory following the simple loading after exhaustion of some trace takes a certain shape of the limit mode, that is, there is a correspondence between the deformation trajectory geometry and the form of response. A new variant of constitutive

Abstract The dynamics of a biaxial four-wheeled vehicle is simulated when drive wheels of one of its axes get into the \(\mu\)-split surface. The \(\mu\)-split surface is a section of the reference plane with different friction coefficients for the left and right wheels of one of the vehicle axes. Depending on the conditions of vehicle motion, the interaction of its wheels with the reference plane

Abstract We consider a system consisting of a platform moving translationally along a fixed straight line in the presence of the viscous friction force and a body making a given translational motion along the same straight line relative to the platform due to internal forces. In relative motion, the value of the body velocity is bounded. It is proved that in the case of linear viscous friction, unlimited

Abstract The article presents new experimental results on the penetration of vertical plane and circular turbulent jets through a free surface of a liquid in relatively narrow channels of different extent with a overflow weir runoff mode. The behavior of the obtained experimental correlations for the period of arising self-oscillating flow modes is discussed. The analysis of a number of discovered

Abstract Dynamic solutions of an analogue of the Prandtl problem in the case of a cylindrical layer, including terms with \(\alpha^{-1}\) and \(\alpha^{0}\), for various configurations of cylinders are obtained and analyzed on the basis of asymptotic analysis with a natural small geometric parameter \(\alpha\) without any static or kinematic hypotheses.

Abstract A closed mathematical model of a double Magnus type wind turbine with a horizontal axis is constructed. The propellers of the turbine are supposed to rotate in opposite directions. For such a system, equations of motion are derived. In the case of dimensions of the front propeller being two times smaller than dimensions of the rear propeller, operating modes and a trapped power coefficient

Abstract In this paper, we consider the equilibrium shapes of a finite volume of liquid on a horizontal rotation surface due to gravity and centrifugal forces under the action of surface tension. It is shown that a solution exits only if the angular velocity is less than the critical value. For large droplets, the angular velocity was obtained analytically.

Abstract For deep bed filtration equations, exact solutions are constructed that describe typical laboratory experiments: injection of a suspension into a porous sample. In the case of kinetic equation with an arbitrary dependence of the clogging rate on the retention (the volume of deposited particles per unit filter volume), the solution is obtained in quadratures. In the special case of linear dependence

Abstract A two-dimensional problem of time-optimal rotation of a mechanical system consisting of a rigid body and a mass point is considered. The mass point interacts with the body by internal forces only. The periodic optimal trajectories of the mass point passing through the rigid body center of inertia are found.

Abstract We classify \(U_{p}\)-sets for the Walsh system depending on the interval of all values of \(p\) for which a given set is a \(U_{p}\)-set.

Abstract The work presents the definition, properties and a method of construction of an attainability set in the neighborhood of a periodic attractor of a nonlinear dynamical system and solves the problem of inverse transition in a bistable system.

Abstract Some issues related to the development of methods for forecasting creep processes are considered. For this purpose, the temporal equivalence principle, the temperature-time analogy, the stress-time analogy, other analogies, and the modern relations between stresses and strains of the heriditary type in the form that allows describing irreversible plastic deformations are used. The expressions

Abstract Exact analytical solutions of diffraction problems are obtained for nonstationary plane and cylindrical waves on a half-plane at following mixed boundary conditions: the Neumann condition on one side of the half-plane and the Dirichlet condition on the other one. The numerical analysis of near-front asymptotic solutions shows that there is a considerably greater attenuation of blast waves

Abstract General properties of the theoretic creep curves for volumetric, longitudinal, and lateral strains generated by the Rabotnov physically nonlinear constitutive equation for non-aging viscoelastic materials under uniaxial loading are studied analytically assuming four material functions of the relation are arbitrary. The expressions for Poisson’s ratio via the strain state parameter and via

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