The problem of limiting equilibrium of a heavy rigid body resting with three points on a horizontal plane with Coulomb friction (tripod) is considered. The body can be brought out of the state of limiting equilibrium by the shift torque of a couple of active forces lying in the support plane, i. e., by the torque whose value does not disturb the equilibrium of the body but, when the torque takes a greater value, equilibrium is broken. The maximum value of the torque of friction forces in the supports, counteracting the shift torque, is found.

The biatomic crystal with CsCl structure is considered with the interatomic interactions described by the $\beta$-Fermi-Pasta-Ulam-Tsingou potential. The case of a large difference in the atomic masses of the components is analyzed when there is a gap in the phonon spectrum of the crystal. A spatially localized large-amplitude vibrational mode, called the discrete breather (DB), is found by applying a localization function to the delocalized nonlinear vibrational mode (DNVM). The DNVM and consequently the DB have frequencies in the gap of the phonon spectrum. It is shown that the DB can be set in motion using a physically motivated ansatz.

In this work, we construct and analyze a mathematical model of coupled longitudinaltransverse vibrations of a rectangular pretensioned strip under conditions of internal combinational resonance between two transverse modes and one longitudinal vibration mode of the structure in view of advanced nanosystems made of two-dimensional materials. The issues of generating frequency combs based on the proposed nanoresonator design are investigated in the context of the development of nondestructive methods of laser optothermal excitation of vibrations of atomically thin nanostructures. Conditions have been analytically found for the amount of deformation of the initial tension of the layer required to realize resonance between eigenmodes with given indices of variability along the length. The necessary relationships between the indices of vibration modes involved in nonlinear interaction are determined. It is shown that, under conditions of internal resonance, beats are excited in the system, the spectrum of which has the form of a frequency comb. Two qualitatively different types of beats are identified — those caused by the initial excitation in the working longitudinal form of vibrations and in two transverse forms. A significant dependence of the spectral composition of the generated frequency combs on the relationships between the amplitudes of the initial disturbance along three interacting modal coordinates and on the value of the internal frequency detuning parameter of the system is shown.

The features of air suspension flow past a cube are investigated. A solution is found to the plane problem of the flow of two-phase medium (water drops with a volume fraction of no more than 1% in the air flow, with a fixed dimension of the dispersed phase on the boundary of supply) past a body with square cross-section. The fields of distribution of physical quantities (pressure, velocity, concentration) at fixed time instants and the dynamics of their variation are presented and analyzed. The local features of the air suspension flow in the near wake are compared with the topology of the one-phase air flow. It is shown that the presence of the second phase in the air suspension leads to turbulization of the flow at small Reynolds numbers and causes the formation of a viscous vortex street in the modes of laminar flow in the subsonic region. An estimate is given of the sufficiency of the distance of the boundaries of the computational region from the cube for the solution of the plane problem of external multiphase aerodynamics.

This article has developed a method for calculating the emissive characteristics of water vapor in the region of $6.3$ $\mu $m under conditions of nonlinear thermal nonequilibrium. A formula is presented for calculating line intensity for various combinations of translational, rotational and vibrational temperatures. When calculating the vibrational energy, the harmonic oscillator model was used. To calculate the rotational energy of H$_2^{}$O, the molecule of which is an asymmetric top, the model of the effective rotational Hamiltonian was used. A solution to the radiation transfer equation is obtained in the absence of scattering, when the medium is capable of both emitting and absorbing radiation. Test calculations were carried out for three temperatures: $600$ K, $1000$ K, and $1550$ K. A comparison of the calculation results with experimental data on the transmittance of a homogeneous H$_2^{}$O layer showed satisfactory agreement. An analysis of the influence of thermal nonequilibrium on the emissive characteristics of a homogeneous H$_2^{}$O layer was carried out for various combinations of translational, rotational and vibrational temperatures for high and low pressure values. It is shown that, in this spectral range, thermal nonequilibrium has a very weak effect on the transmittance of the layer, but very strongly affects the nonequilibrium Planck function, which behaves significantly nonlinearly with respect to characteristic temperatures and cannot be described by any of the temperatures of the energy modes: translational, rotational and vibrational. The values of the nonlinear non-equilibrium Planck function in the range $1000$–$1800$ cm$^{-1}$ are closest to the radiation of the black body at a translational temperature that coincides with the value of the vibrational temperature of the second energy mode — $T_{v2}^{}$. This is due to the fact that in the region of $6.3$ $\mu $m the main mechanism of radiation generation is the spontaneous deactivation of the H$_2^{}$O deformation mode. Accordingly, the influence of thermal nonequilibrium on the spectral energy brightness is great. This important result makes it possible to significantly simplify the calculation of emission characteristics when using such simplified approximate methods as the statistical model of the band, the $k$-distribution method since, in fact, the databases created for these techniques only need to take into account the effect of disequilibrium on the Planck function.

In this paper we consider the steady inhomogeneous shear flow of a viscous incompressible fluid taking into account the possibility of solid-body rotation of a representative volume. Mathematically, the contribution of couple stresses manifests itself in an increase in the order of the system of governing differential equations. We discuss problems of the existence of an exact solution within the framework of the class of functions linear in some of the coordinates. It is shown that the problem of overdetermination of the system of equations, which is traditional for models describing shear flows, does not arise for the chosen class of solutions. An exact solution is constructed for the velocity field of the flow. Also, an exact solution of the boundary-value problem describing adhesion and superadhesion on the boundaries of the flow region is analyzed in dimensionless form. It is shown that these exact solutions are capable of describing stagnation regions observed in real fluids and the effect of increase in velocities.

The Poincaré – Zhukovsky – Hough model describing the motion of a rigid body with an ellipsoidal cavity filled with an ideal vortex liquid is used. The possibility of regular precession in a uniform force field of a system not possessing axial symmetry is shown. For the case where the axis of proper rotation is one of the system principal inertia axes and the center of gravity lies on this axis, two conditions of precession are obtained. One of the conditions coincides with the condition of regular precession in the absence of external forces for the system without axial symmetry found earlier by the author. This condition imposes one constraint on the system configuration. The other condition relates the proper rotation and precession velocities to the mechanical parameters of the system. A record is given of the conditions in the form of relations between the inertia moments of the rigid shell and the semiaxes of the ellipsoidal cavity, as well as between the distance to the center of gravity and the nutation angle, the precession velocity and the proper rotation velocity. It is shown that in the case where the cavity differs little from the sphere, the conditions obtained differ from the Lagrange conditions for an axisymmetric rigid body with a fixed point in a uniform gravity field by small values of the second order.

The interaction of various climatic zones (northern, temperate and southern) as circulation cells of large-scale atmospheric currents is represented in the form of a superposition of six-dimensional systems that describe the motion of stratified fluids in the space of linear velocity and temperature fields. The effect of each of the zones on neighboring zones occurs only through temperature gradients along the meridional direction which are due to sources and sinks of heat (short-wave radiation, downcoming long-wave radiation, outgoing long-wave radiation). The resulting nonlinear system, in which each of the blocks contains vortex and temperature components of the fields, is discretized by an implicit scheme. Long-period nonlinear oscillations are modeled, showing the natural “breath” of the climate of the atmosphere, in which the amplitudes of vortex intensity and temperature differences of various zones change with characteristic times of decades. In this case, the transition between quasi-stationary states of the system can occur over several years. A comparison is made between various numerical methods that show long-term oscillations in convective systems.

In this paper, we examine a specific class of quadratic operators. For these operators, we identify all fixed points and categorized their types in the general case. Our analysis reveals that there are no attractive fixed points except the origin. Additionally, we investigate the global dynamics in the two-dimensional case and generalize several results obtained for lowerdimensional scenarios.

In this work, the interaction of the kink in the $\varphi^4$ model with two point impurities is considered. A point impurity is described using the Dirac delta-function. The case of an attractive impurity is analyzed. It is shown that the interaction of the kink with the impurities leads to the excitation of long-lived small-amplitude breather-type waves localized on them. Their structure and associated dynamics have been investigated analytically and numerically. Using the collective variable method, a system of two differential equations describing the coupled dynamics of the waves localized on the impurities has been obtained. This system of equations has solutions: in the form of in-phase oscillations, if the initial amplitudes of the waves localized on the impurities are equal; and in the form of antiphase oscillations, if one of the initial amplitudes is zero. In all other cases of initial amplitudes, the system has solutions in the form of beats. Numerically, using the method of lines, coupled in-phase oscillations, antiphase oscillations, and beats of the waves localized on the impurities were also obtained. The oscillations of the waves localized on the impurities are accompanied by radiation. The existence of two possible oscillation frequencies was found, both analytically and numerically. It is shown that these frequencies do not depend on the initial kink velocity but strongly depend on the distance between the impurities. As the distance between the impurities increases, the frequencies merge into one — the frequency obtained for the case of a single impurity. The dependencies of the frequencies on the distance between the impurities, found numerically and analytically, agree well for large distances, when the interaction between the impurities is weak, and begin to differ noticeably at small distances, when the interaction between the impurities is strong. The analytical values of the obtained frequencies are always greater than the numerical ones.

This paper contains an analysis of the problems concerning the control of underactuated systems. As an underactuated system an inverted pendulum on a wheel system is chosen since it has one motor that affects both the motion of the wheel and the angular position of the pendulum. The first objective of this work is to develop a motion algorithm bringing the system from the initial to the final point while both points are connected with a straight line and the system starts motion from the equilibrium position. The second objective is to study applicability of the modal control technique to the system and observe the range of its applicability in the case of parametric uncertainties in the system.
The proposed motion trajectory is built on the basis of the maximum allowed angular velocity of the motor and linear optimization techniques. The modal controller is applied to the initial parametric configuration of the system and to the system with a significant degree of parametric uncertainty. The controller demonstrates high robustness to constant parametric uncertainty expressed by the stability of the trajectory tracking process and a wide range of applicability.