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A dynamical system d X/d t = F(X; A) is treated where F(X; A) is a polynomial (or some general type of radical contained) function in the vectors of state variables X ∈ ℝn and parameters A ∈ ℝm. We are looking for stability domains in both spaces, i.e. (a) domain ℙ ⊂ ℝm such that for any parameter vector specialization A ∈ ℙ, there exists a stable equilibrium for the dynamical system, and (b) domain 𝕊 ⊂ ℝn such that any point X* ∈ 𝕊 could be made a stable equilibrium by a suitable specialization of the parameter vector A. [ABSTRACT FROM AUTHOR]

In the present study, a theoretical model of state-resolved transport properties in electronically excited atomic species developed earlier is applied to argon and carbon atomic species. It is shown that for Ar and C, similarly to the case of atomic nitrogen and oxygen, the Slater-like models can be applied to calculate diameters of electronically excited atoms. Using the Slater-like model it is shown that for half-filled N ( 2 p x 1 p y 1 p z 1 ) and full-filled Ar ( 3 p x 2 p y 2 p z 2 ) electronic shells the growth of atomic radius goes slowly compared to C ( 2 p x 1 p y 1 ) and O ( 2 p x 2 p y 1 p z 1 ). The effect of collision diameters on the transport properties of Ar and C is evaluated. The influence of accounted number of electronic levels on the transport coefficients is examined for the case of Boltzmann distributions over electronic energy levels. It is emphasized that in the temperature range 1000–14000 K, for Boltzmann-like distributions over electronic states the number of accounted electronic levels do not influence the transport coefficients. Contrary to this, for higher temperatures T > 14000 K this effect becomes of importance, especially for argon. [ABSTRACT FROM AUTHOR]

The software package Planet Atmosphere Investigator of Non-equilibrium Thermodynamics (PAINeT) has been devel-oped for studying the non-equilibrium effects associated with electronic excitation, chemical reactions and ionization. These studies are necessary for modeling process in shock tubes, in high enthalpy flows, in nozzles or jet engines, in combustion and explosion processes, in modern plasma-chemical and laser technologies. The advantages and possibilities of the package implementation are stated. Within the framework of the package implementation, based on kinetic theory approximations (one-temperature and state-to-state approaches), calculations are carried out, and the limits of applicability of a simplified description of shock-heated air flows and any other mixtures chosen by the user are given. Using kinetic theory algorithms, a numerical calculation of the heat fluxes and relaxation terms can be performed, which is necessary for further comparison of engineering simulation with experi-mental data. The influence of state-to-state distributions over electronic energy levels on the coefficients of thermal conductivity, diffusion, heat fluxes and diffusion velocities of the components of various gas mixtures behind shock waves is studied. Using the software package the accuracy of different approximations of the kinetic theory of gases is estimated. As an example state-resolved atomic ionized mixture of N/N+/O/O+/e− is considered. It is shown that state-resolved diffusion coefficients of neutral and ionized species vary from level to level. Comparing results of engineering applications with those given by PAINeT, recommendations for adequate models selection are proposed. [ABSTRACT FROM AUTHOR]

Bileaflet mechanical valves are most commonly used for heart valve replacement. Nowadays swirling blood flow is registered in different parts of the cardiovascular system: left ventricle, aorta, arteries and veins. In present contribution for the first time the physiological swirling flow inlet conditions are used for numerical simulation of aortic bileaflet mechanical heart valve hemodynamics. Steady 3-dimensional continuity and RANS equations are employed to describe blood motion. The Menter SST model is used to simulate turbulence effects. Boundary conditions are corresponded to systolic peak flow. The domain was discretized into hybrid tetrahedral and hexahedral mesh with an emphasis on wall boundary layer. A system of equations was solved in Ansys Fluent finite-volume package. Noticeable changes in the flow structure caused by inlet swirl are shown. The swirling flow interaction with the valve leaflets is analyzed. A central orifice jet changes its cross-section shape, which leads to redistribution of wall shear stress on the leaflets. Transvalvular pressure gradient and area-averaged leaflet wall shear stress increase. Physiological swirl intensity noticeably reduces downstream of the valve. [ABSTRACT FROM AUTHOR]

Research is conducted into variation in the stress-strain state of the corneoscleral shell of the human eye under loading by a flat base stamp of varying weight. A three-dimensional finite-element model of the contact problem of loading of the corneoscleral shell in the ANSYS program package is presented. Cornea and sclera are modeled as conjugated transversely isotropic spherical shells. The cornea is modeled as a multilayer shell with variable thickness in which all modeled layers have their own individual elastic properties. The research deals with the numerical calculation of the diameter of the contact zone between the shell and the stamp. Values of correction coefficients for intraocular pressure are obtained depending on the thickness of the corneal shell in its center, allowing the true intraocular pressure to be determined more accurately. [ABSTRACT FROM AUTHOR]

The paper identifies the importance of the Leonhard Euler’s discoveries in the field of shipbuilding for the scientific evolution of academician A. N. Krylov and for the modern knowledge in survivability and safety of ships. The works by Leonard Euler “Marine Science” and “The Moon Motion New Theory” are discussed. [ABSTRACT FROM AUTHOR]

Isaak Pavlovich Ginzburg (1910-1979) was a professor at the hydroaeromechanics department of Leningrad State University, a prominent scientist, an outstanding organizer and a brilliant educator, who had trained more than one generation of specialists in the field of fluid, gas and plasma mechanics. Many of his students became major scientists and organizers of science. The present paper is devoted to the students of I.P. Ginzburg graduated from the Mathematics and Mechanics Faculty of Leningrad State University. [ABSTRACT FROM AUTHOR]

Franz Reuleaux (1829 - 1905) is a famous mechanical engineer, a Professor of the Berlin Royal Technical Academy. He became widely known as an engineer-scientist, a Professor and industrial consultant, education reformer and leader of the technical elite of Germany. He directed the design and manufacture of over 300 models of simple mechanisms. They were sold to many famous universities for pedagogical and scientific purposes. Today, the most complete set is at Cornell University, College of Engineering. In this article we discuss the history, the modern state and our using the Reuleaux models that survived at St. Petersburg State University for educational purposes. We present description of certain models and our electronic resource with these models. We provide the information of similar electronic resources from other universities. [ABSTRACT FROM AUTHOR]

An experiment on the measurement of the ratio of the axial coupling constant to the vector one is under development. The main idea of the experiment is to measure the values of A and B in the same setup. An additional measurement of the polarization is not necessary. The accuracy achieved to date in measuring λ is 2 × 10-3. It is expected that in the experiment the accuracy will be of the order of 10-4. Some particular problems of mathematical modeling concerning the experiment on the measurement of the ratio of the axial coupling constant to the vector one are considered. The force lines for the given tabular field of a magnetic trap are studied. The dependences of the longitudinal and transverse field non-uniformity coefficients on the coordinates are regarded. A special computational algorithm based on the law of a charged particle motion along a local magnetic force line is performed for the calculation of the electrons and protons motion time as well as for the evaluation of the total number of electrons colliding with the detector surface. The average values of the cosines of the angles with the coefficients of a, A and B have been estimated. [ABSTRACT FROM AUTHOR]

Results of the experimentation on the destruction of the phase change materials (beeswax and paraffin) by the electric explosion of conductors are presented. The process of the explosion of copper and nickel titanium wires in both pure PCM and its mixture with nonosized additives of cuprous oxide is analyzed. The effect of this additive on the process of the expansion of the electric-discharge plasma during the electric explosion of conductors and on the strength of composite materials is demonstrated. The piezoprobe-based method of measurement of the radial pressure during samples destruction is developed. The experiments made it possible to determine the dimensions of the melting channel formed inside the samples during the explosion and the subsequent expansion of the electric-discharge plasma. The experiments are performed on the generator of short-term high-voltage pulses capable to shape the voltage of (10-24) kV. [ABSTRACT FROM AUTHOR]

Results of study of deformation and fracture of ring-shaped samples made of thin strips of cuprum, aluminum and steel in wide range of loading velocity are presented. Three developed by us schemes of magnetic-pulse method are used for the samples loading. The method of samples fracture with the high electrical resistance (e.g. steel) is proposed. Crack velocity at the sample fracture is estimated. Fracture surfaces are inspected. Mechanisms of dynamic fracture of the sample arere discussed. [ABSTRACT FROM AUTHOR]

The resistive sausage instability of the relativistic electron beam in dense gas-plasma medium in the case of the generation of equilibrium return plasma current is investigated. In this situation the eigenvalue equation of this instability is obtained. The stabilizing and destabilizing effects of the phase mixing and generation of the return plasma current respectively have been shown. [ABSTRACT FROM AUTHOR]

In the paper we continue the study of precipitation regions of high-energy charged particles, carried out by the authors since 2002. In contrast to previous papers, where a stationary source of electrons was considered, it is assumed that the source moves along a low circular near-earth orbit with a constant velocity. The orbit position is set by the inclination angle of the orbital plane to the equatorial plane and the longitude of the ascending node. The total number of injected electrons is determined by the source strength and the number of complete revolutions that the source makes along the circumference. Construction of precipitation regions is produced using the computational algorithm based on solving of the system of ordinary differential equations. The features of the precipitation regions structure for the dipole approximation of the geomagnetic field and the symmetrical arrangement of the orbit relative to the equator are noted. The dependencies of the precipitation regions on different orbital parametres such as the incline angle, the ascending node position and kinetic energy of injected particles have been considered. [ABSTRACT FROM AUTHOR]

The presented model of thermodynamic evolution of an open system far from equilibrium is based on the modern results of nonequilibrium statistical mechanics, the nonlocal theory of nonequilibrium transport developed by the author and the Speed Gradient principle introduced in the theory of adaptive control. Transition to a description of the system internal structure evolution at the mesoscopic level allows a new insight at the stability problem of non-equilibrium processes. The new model is used in a number of specific tasks. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of determining the structure of the electric field near the surface of a flat insulated body under conditions of a deep vacuum. It is assumed that the emitted particles are electrons leaving the body surface under the influence of ionizing radiation whose velocities distribution near the surface is isotropic. It is estimated the thickness of the screening layer under conditions of stationary emission from a flat surface. The solutio of the problem of determining a stationary self-consistent electric field near the surface is found in a simple analytical form. The thickness of the screening layer is calculated from this formula. [ABSTRACT FROM AUTHOR]

In this paper we study the structure of near-surface layers of artificial and natural cosmic bodies, electric fields screening features, and the influence of the surface curvature on the charge state of bodies and their surfaces. The estimates of the thickness of the screening layer are obtained for the case of pulsed irradiation by a monochromatic beam related to the ultraviolet part of the spectrum. The problem of determining a stationary self-consistent electric field for a monoenergetic, monodirectional electron beam emitted from a plane body surface is solved analytically. It is shown that the thickness of the screening layer is finite and, in order, is equal to the Debye radius. [ABSTRACT FROM AUTHOR]

The effect of spatial inhomogeneity of the effective cornea stiffness distribution on the mechanical properties of the eye is investigated on the basis of the two-component model of the eyeball, in which the cornea is represented by a momentless deformable, linearly elastic surface and the scleral region by an elastic element that responds to changes in intraocular pressure by changes in volume. The approach used makes it possible to consider within the same model both the natural corneal inhomogeneity and mechanical consequences of local cornea weakening owing to surgical procedures. The dependences on changes in intraocular pressure of parameters that characterize deformation properties of both the cornea (apex displacement) and the eyeball as a whole (change in intraocular volume) are obtained. For moderate inhomogeneity they differ from the same dependences for the homogenous cornea with effective stiffness equal to the average value for the corresponding inhomogeneous distribution only slightly. However, if the effective stiffness amplitude is very high, corneal inhomogeneity discernibly affects the integral response of the cornea and the eyeball as a whole to changes in pressure. The effect of inhomogeneity on the data of tonometry also mainly depends on the average effective corneal stiffness. The difference between the tonometric and true pressures increases with surgical cornea weakening in the apical region for both Schiøtz and Maklakoff tonometers. [ABSTRACT FROM AUTHOR]

Stomata are the structural elements of plant epidermis which control transpiration and gas exchange. Each stoma consists of two guard cells divided by the stomatal aperture. These cells are capable of reversible deformations determining the width of aperture. It is known that these deformations depend on the value of turgor pressure in the guard cells and on the structure of their walls. In this work, the influence of the outer tangential wall geometry of the guard cells on stomatal movements is estimated by means of the finite element method in the ANSYS software. The application of modelling has shown that cuticular outgrowths on the tangential walls influence the degree and pattern of guard cell deformations. The outgrowths prevent wide opening of the stomatal aperture and cause its sinking deep into leaf epidermis. The functional significance of such stomatal movements is discussed. It is deduced that the discovered phenomenon had great importance to the survival of laurophyllous plants in conditions of aridization. [ABSTRACT FROM AUTHOR]

The present contribution is aimed at numerical and experimental study of inlet swirling flow in a model of the carotid artery. Flow visualization is performed both with the ultrasound color Doppler imaging mode and with CFD data postprocessing of swirling flows in a carotid artery model. Special attention is paid to obtaining data for the secondary motion in the internal carotid artery. Principal errors of the measurement technique developed are estimated using the results of flow calculations. [ABSTRACT FROM AUTHOR]

Nonlinear dynamics of two electronic oscillators coupled via a memristive device has been investigated. Such model mimics the interaction between synaptically coupled brain neurons with the memristive device imitating neuron axon. The synaptic connection is provided by the adaptive behavior of memristive device that changes its resistance under the action of spike-like activity. Mathematical model of such a memristive interface has been developed to describe and predict the experimentally observed regularities of forced synchronization of neuron-like oscillators. [ABSTRACT FROM AUTHOR]

A statically unbalanced rotor in viscoelastic orthotropic supports equipped with an automatic ball balancer (ABB), the axis of symmetry of which does not coincide with the symmetry axis of the rotor, is considered. Based on an analysis of the equations describing the stationary modes of motion of the system, the principal impossibility of complete balancing of the rotor is shown. The possibility of the existence of two types of stationary modes is established, one of which has a constant average amplitude of residual vibration equal to the eccentricity of the ABB. The solution corresponding to this almost balanced mode is constructed analytically. A study is made of its asymptotic stability. [ABSTRACT FROM AUTHOR]

A finite-element model is presented for a healthy person and for a person with an injury in this section of the spine. The mechanical parameters of the bone tissue of the vertebrae, intervertebral discs, arcuate joints, and ligaments, are modeled on the basis of data from literature sources. Elements of the transitional thoracolumbar spine are considered as isotropic, homogeneous and linearly elastic material. The obtained models allow for a comparative analysis of the spine of a healthy person and the presence of injure in the transitional thoracolumbar spine. [ABSTRACT FROM AUTHOR]

The mathematical model of measuring of the intraocular pressure (IOP) by Maklakoff tonometer in the software package ANSYS, Inc. is presented. The dependence of the tonometric pressure on the weight of the tonometer before and after LASIK and Femto LASIK surgical corrections of hyperopia is analized. Cornea-scleral eye shell is modeled as two thin transversely isotropic joint spherical shells with different radii of curvature and biomechanical properties. To analyze the biomechanical state of the corneoscleral shell of the eye after the surgery the elastotonometry method is used. Firstly sequential measurements of intraocular pressure by Maklakoff tonometers weighting 5; 7.5; 10 and 15 g is carried out, then the dependence curve of tonometric IOP vs. tonometer weight is plotted and analyzed. The results of the calculations are compared with the clinical data. [ABSTRACT FROM AUTHOR]

The paper describes a new scenario for the transition to complex dynamics in a vibrating system with an unbalanced rotor and a relay feedback control. We show that the transition from a regular dynamics without switching events in the relay element to an irregular dynamics which takes place completely in the hysteresis region occurs via a cascade of persistence border collisions. [ABSTRACT FROM AUTHOR]

The Lagrange-Maxwell equations for the pendulum in the form of a conductive frame, which is suspended in a uniform sinusoidal electromagnetic field of constant intensity, are obtained. The procedure for obtaining simplified mathematical models by a traditional method of separating fast and slow motions with subsiquent averaging a fast time is used. It is shown that this traditional approach may lead to inappropriate mathematical models. Suggested ways on how this can be avoided for the case are considered. The main statements by numerical experiments are illustrated. [ABSTRACT FROM AUTHOR]

Some classes of switched mechanical systems with dissipative and potential forces are considered. The case, where either dissipative or potential forces are essentially nonlinear, is studied. It is assumed that the zero equilibrium position of the system is asymptotically stable at least for one operating mode. We will look for sufficient conditions which guarantee the preservation of asymptotic stability of the equilibrium position under the switching of modes. The Lyapunov direct method is used. A Lyapunov function for considered system is constructed, which satisfies the differential inequality of special form for every operating mode. This inequality is nonlinear for the chosen mode with asymptotically stable equilibrium position, and it is linear for the rest modes. The correlations between the intervals of activity of the pointed mode and the intervals of activity of the rest modes are obtained which guarantee the required properties. [ABSTRACT FROM AUTHOR]

The problem of finding optimal temperature control of the material body state under the unknown in advance parameters of the external medium is formalized and studied in this paper. The problems of this type arise frequently in the real life. An optimal thermal regime is necessary to apply at the soil thawing or freezing, drying the building materials, heating the concrete to obtain the required strength, and so on. Problems of such type one can analyze making use the apparatus and methods of game theory. For describing the influence of external medium on the characteristics of different materials we make use the many-step two person zero-sum game in this paper. The compromise solution is taken as the optimality principle. The numerical example is given. [ABSTRACT FROM AUTHOR]

A method of positional and force-torque control of robots is proposed. Prototypes of the system and the master handle have been created. Algorithm of bias estimation and gravity compensation for force-torque sensor and force-torque trajectory correction are described. [ABSTRACT FROM AUTHOR]

The problem of the one-wheeled crew motion on smooth horizontal ice is considered. The motion equations are worked out in quasicoordinates in the form of Euler-Lagrange’s equations. The variety of stationary motions is defined. Stability of some stationary motions is investigated. Comparison of the results received for a similar model of one-wheeled crew at its motion on the horizontal plane without slipping is carried out. [ABSTRACT FROM AUTHOR]

In this article the mathematical model of a planar space robot manipulator is under study. The space robot manipulator represents a solid body with attached manipulators. The system of equations of motion is determined using the Lagrange’s equations. The control problem concerning moving the robot to a given point and return it to a given trajectory in the phase space is solved. Changes of generalized coordinates and necessary control actions are plotted for a specific model. [ABSTRACT FROM AUTHOR]

The article presents analysis on construction of two-stage remote control for space robots. This control ensures efficiency of the robot control system at large delays in transmission of control signals from the ground control center to the local control system of the space robot. The conditions for control stability of and high transparency are found. [ABSTRACT FROM AUTHOR]

Motion of a pendulum with damping and vibrating axis of suspension is considered at unconventional values of parameters. Case when the frequency of external loading and the natural frequency of the pendulum in the absence of this loading are of the same order is studied. Vibration intensity is assumed to be relatively low. In this case, the corresponding equation of the pendulum’s motions doesn’t involve an explicit small parameter. To solve the equation a new modification of the method of direct separation of motions is used. As the result, stability conditions of the pendulum inverted position are determined. Effects of damping on these conditions are discussed. [ABSTRACT FROM AUTHOR]

The construction of Lyapunov-Krasovskii functionals for linear time-delay system depends on so-called Lya-punov matrix. We study the problem of finding this matrix by solving the delay-free equations with specific boundary conditions. The special boundary conditions for the delay-free problem are found and it is shown, that the problem of Lyapunov matrix’ finding as a solution of the boundary value problem with delay and the delay-free problem are equivalent. [ABSTRACT FROM AUTHOR]

The goal of the study is to validate Large Eddy Simulation (LES) data on mixing ventilation in an isothermal room at conditions of benchmark experiments by Hurnik et al. (2015). The focus is on the accuracy of the mean and rms velocity fields prediction in the quasi-free jet zone of the room with 3D jet supplied from a sidewall rectangular diffuser. Calculations were carried out using the ANSYS Fluent 16.2 software with an algebraic wall-modeled LES subgrid-scale model. CFD results on the mean velocity vector are compared with the Laser Doppler Anemometry data. The difference between the mean velocity vector and the mean air speed in the jet zone, both LES-computed, is presented and discussed. [ABSTRACT FROM AUTHOR]

The study of the film flow occurred under the influence of a gas slug flow is of definite interest in heat and mass transfer during the motion of a coolant in the second circuit of a nuclear water-water reactor. Thermohydraulic codes are usually used for analysis of the such problems in which the motion of the liquid film and the vapor is modeled on the basis of a one-dimensional balance equations. Due to a greater inertia of the liquid film motion, film flow parameters changes with a relaxation compared with gas flow. We consider a model problem of film flow under the influence of friction from gas slug flow neglecting such effects as wave formation, droplet breakage and deposition on the film surface, evaporation and condensation. Such a problem is analogous to the well-known problems of Couette and Stokes flows. An analytical solution has been obtained for laminar flow. Numerical RANS-based simulation of turbulent flow was performed using OpenFOAM. It is established that the relaxation process is almost self-similar. This fact opens a possibility of obtaining valuable correlations for the relaxation time. [ABSTRACT FROM AUTHOR]

The flows of CO2/CO/O2/O/C and CO2/CO/O mixtures behind shock waves are studied in the three-temperature, two-temperature and one-temperature approximations. The influence of the vibrational relaxation and chemical reactions on the flow composition, temperature and velocity is investigated. It is shown that the vibrational non-equilibrium has a significant effect on the macroscopic parameters of the flow near the front of the shock wave. It was found that the composition of the mixture has the greatest effect on the numerical density of CO molecules and O atoms. Also, significant differences between the values of the vibrational temperature of the asymmetric regime have been revealed. [ABSTRACT FROM AUTHOR]

The motion of a fully balanced Jeffcott rotor excited by the external torque is considered. The mechanical model of the rotor takes into account the orthotropy of visco-elastic characteristics of the shaft and the influence of viscous external and internal damping forces. The self-excited whirling motions, due to the loss of stability of proper rotation, are investigated. It is established that, for sufficiently strong orthotropy of the shaft, there are two regions of instability of rotor’s basic motion. In the first region the steady-state motion of the rotor is regular synchronous whirling, and in the second one there are asynchronous self-excited vibrations. We obtained the analytic formulas for the threshold values of the torque, which limit existence regions of both whirling modes. [ABSTRACT FROM AUTHOR]

This paper is focused on the problem of control of plane motions of a space tethered system (STS). The STS is modeled as a heavy rod with two point masses. Point masses are fixed on the rod. A third point mass can move along the rod. The control is realized as a continuous change of the distance from the centre of mass of the tethered system to the movable mass. New limited control laws processes of excitation and damping are built. Diametric reorientation and gravitational stabilization to the local vertical of an STS were obtained. The problem is solved by the method of Lyapunov’s functions of the classical theory of stability. The theoretical results are confirmed by numerical calculations. [ABSTRACT FROM AUTHOR]

The problem of natural stabilization of a coupled periodical system is solved by determining couplings that ensure both existence and asymptotic stability of an oscillation. The developped approach consists in reducing the initial subsystems such that they comprise cycles, and simultaneously, in coupling the subsystems into the system that admits an symptotically stable periodic motion. The couplings are supposed to be smooth periodic in time functions. [ABSTRACT FROM AUTHOR]

In this paper we consider an autonomous mechanical system in a finite neighborhood of the zero of the phase space of states. The system is given as a matrix differential equation in the Cauchy form with the right-hand side of the polynomial structure. We propose a method for constructing a sequence of linear inhomogeneous differential inequalities for Lyapunov functions. As a result, we obtain estimates of transient processes in the form of functional inequalities. [ABSTRACT FROM AUTHOR]

The paper deals with first-order differential equations under the assumption that the right-hand side is a periodic function of time and continuous in the set of arguments. Pliss V.A. obtained the first results for a particular class of equations and showed that a number of theorems can not be continued. In this paper, it was possible to reduce the restrictions on the degree of smoothness of the right-hand side of the equation and obtain upper and lower bounds on the number of possible periodic solutions. [ABSTRACT FROM AUTHOR]

An important requirement for the process of solving differential equations in Dynamics, such as the equations of the motion of celestial bodies and, in particular, the motion of cosmic robotic systems is high accuracy at large time intervals. One of effective tools for obtaining such solutions is the Taylor series method. In this connection, we note that it is very advantageous to reduce the given equations of Dynamics to systems with polynomial (in unknowns) right-hand sides. This allows us to obtain effective algorithms for finding the Taylor coefficients, a priori error estimates at each step of integration, and an optimal choice of the order of the approximation used. In the paper, these questions are discussed and appropriate algorithms are considered. [ABSTRACT FROM AUTHOR]

In this paper the problem of damping of the linear systems oscillations with piece-wise constant control is solved. The motion of bar construction is reduced to the form described by Hill’s differential equation using the Bubnov-Galerkin method. To calculate switching moments of the one-side control the method of sequential linear programming is used. The elements of the fundamental matrix of the Hill’s equation are approximated by trigonometric series. Examples of the optimal control of the systems for various initial conditions and different number of control stages have been calculated. The corresponding phase trajectories and transient processes are represented. [ABSTRACT FROM AUTHOR]

The optimal controlled motion of a mechanical system, that is determined by the linear system ODE with constant coefficients and piecewise constant control components, is considered. The number of control switching points and the heights of control steps are considered as preset. The optimized functional is combination of classical time criteria and “Expenditure criteria”, that is equal to the total area of all steps of all control components. In the absence of control, the solution of the system is equal to the sum of components (frequency components) corresponding to different eigenvalues of the matrix of the ODE system. Admissible controls are those that turn to zero (at a non predetermined time moment) the previously chosen frequency components of the solution. An algorithm for the finding of control switching points, based on the necessary minimum conditions for mixed criteria, is proposed. [ABSTRACT FROM AUTHOR]

A mechanical system describing by the second order linear differential equations with a positive parameter at the velocity forces and with time delay in the positional forces is studied. Using the decomposition method and Lyapunov–Krasovskii functionals, conditions are obtained under which from the asymptotic stability of two auxiliary first order subsystems it follows that, for sufficiently large values of the parameter, the original system is also asymptotically stable. Moreover, it is shown that the proposed approach can be applied to the stability investigation of linear gyroscopic systems with switched positional forces. [ABSTRACT FROM AUTHOR]

The problem of monoaxial stabilization of a rigid body is studied. It is assumed that a linear time-invariant dissipative torque and a time-varying restoring torque vanishing as time increases act on the body. Both the case of linear restoring torque and that of essentially nonlinear one are considered. With the aid of the decomposition method, conditions are obtained under which we can guarantee the asymptotic stability of an equilibrium position of the body despite the vanishing of the restoring torque. A numerical simulation is provided to demonstrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

The work is devoted to the study of the heat generation process in the problem of the dynamics of oscillations of a one-dimensional chain simulating heat formation in an elastic continuous medium under mechanical influences. Formulas for estimating the effect of thermoelasticity are obtained and an analogy is made with the energy of damped oscillations of an anharmonic oscillator. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of constructing equations describing the states of dissipative dynamical systems (media with absorption or damping). The approaches of Lagrange and Hamilton are discussed. A non-symplectic extension of the Poisson brackets is formulated. The application of the Hamiltonian formalism here makes it possible to obtain explicit equations for the dynamics of a nonlinear elastic system with damping and a one-dimensional continuous medium with internal friction. [ABSTRACT FROM AUTHOR]

Free vibrations of an elastic non-circular cylindrical shell of intermediate length are considered. The shell is assumed heterogeneous in the thickness direction, in its part it may be multilayered. In order to derive the equations of stability, we use the Timoshenko–Reissner model. According to it, a shell that is heterogeneous can be replaced by a homogeneous shell with the equivalent bending and transversal shear stiffness. We obtain the approximate asymptotic formula for a frequency that takes into account an influence of a transversal shear and a variability of a directrix curvature. As an example, a three-layer elliptical shell with hinged edges and a soft middle layer is analyzed. [ABSTRACT FROM AUTHOR]

A geometric non-linear endochronic theory of inelasticity in tensor parametric form is considered. In the framework of this theory, the creep strains are modelled. The effect of various schemes of applying stresses and changing of material properties on the development of creep strains is studied. The constitutive equations of the model are represented by non-linear systems of ordinary differential equations which are solved in MATLAB environment by implicit difference method. Presented results demonstrate a good qualitative agreement of theoretical data and experimental observations including the description of the tertiary creep and pre-fracture of materials. [ABSTRACT FROM AUTHOR]