Vol 88, No 4 (2024)

The nonlinear dynamics of a point that remains throughout its motion on the inner part of an absolutely smooth surface of a fixed triaxial ellipsoid is studied. The motion occurs in a uniform field of gravity, the largest of the axes of the ellipsoid is directed along the vertical. The main attention is paid to the motions of the point near its stable equilibrium position at the lowest point of the ellipsoid‘s surface lying on its vertical axis. A qualitative description of conditionally periodic oscillations of the point is given, and an estimate of the measure of the set of initial conditions corresponding to these oscillations is defined. In the resonant case, when the ratio of the frequencies of small linear oscillations is equal to two, the periodic motions of the point are studied; the question of their existence, stability and geometric representation is considered.

The possibility of reversibility of the principle of release from connections, widely used in solving problems of mechanics, is studied. The opposite position is formulated, according to which the movement of the system will not change if the forces acting on it are ignored and connections are imposed, the reactions of which provide the initial movement. In this case, the studied mechanical system is obtained from another, with a large number of degrees of freedom, on which both holonomic ideal connections and nonholonomic ones are superimposed, and movement occurs in the absence of active active forces. The main task is to determine the equations of relations in an expanded space of configurations that uniquely generate given force fields in the original space.

A general solution to the problem of stochastic estimation of navigation parameters of mast antennas of radio engineering complexes (RTCs) located on mobile objects is considered. It is shown that the existing methods for determining navigation parameters using measurements of satellite navigation systems or inertial orientation systems do not provide the required accuracy for solving this problem for such a class of antennas under the action of random disturbances on an object and/or mast.

In this regard, an algorithm is proposed for stochastic estimation of the navigation parameters of a mast antenna of a radio engineering complex located on a mobile object, invariant both to the nature of the movement of the mast and to the nature of the movement of the object. It is shown that this algorithm makes it possible to ensure stability and the required accuracy of estimation under the most general assumptions about the nature of interference of sensitive elements (CE) using a strapless inertial orientation system (BIS). To solve the problem, in the most general case, the BISO includes two groups of CES consisting of three orthogonal accelerometers and three angular velocity sensors (ARC) located, respectively, in the centers of mass of the object and the antenna.

The vectors of the Rodrigue–Hamilton parameters are used as the observed vectors of the navigation parameters of the antenna and the object, and the vector of the DUS output signals located in the center of mass of the antenna is used as their observer. Based on stochastic nonlinear equations of their state vectors and equations of stochastic models of DUS output signals constructed for the most general case of antenna and object motion, a generalized Kalman filter was formed, providing a general solution to the problem of estimating the navigation parameters of a mast antenna of arbitrary design placed on a moving object.

The presented results of numerical modeling allow us to conclude that the proposed approach can be used to solve the problem of high-precision determination of navigation parameters of mast antennas of radio engineering complexes located on mobile objects, using medium and high-precision BIS without correction over a long period of time.

The problem of calculating the equilibrium axisymmetric shape of a liquid drop resting on a non-deformable horizontal plane is formulated. For the first time, an equation for the balance of forces acting on a drop in the vertical direction has been obtained, which completes the formulation of the problem under consideration. A high-precision numerical method for solving the formulated nonlinear problem has been developed. The dependence of the wetting angles of drops on variation of the input data of the problem: the chemical composition of the drop, gas pressure, and the strength of additional weak interaction (for example, van der Waals or electrochemical origin) is studied. For drops of small diameters, the possibility of the existence of two solutions is shown, which correspond to significantly different contact angles: in the first solution, the contact angles are less than 90°, and in the second, they are greater than 90°, reaching values of 160° and more. The existence of two equilibrium forms of a small-diameter drop is confirmed by full-scale experiments. Equilibrium forms of droplets of large diameters can exist only in the presence of an additional weak repulsive force between the liquid and the supporting surface, having an intensity of the order of 10^–7…10^–5 Pa. In this case, for drops of large diameters, there is only one solution.

The problem is considered for the indentation of a stratified elastic strip by a rigid punch of finite dimension with a surface microrelief. Boundary variational formulations of the problem are given using the Poincaré-Steklov operator that maps normal stresses to normal displacements. To approximate this operator the discrete Fourier transform is used. The fast Fourier transform algorithms are applied for numerical realization. A variational formulation of a boundary value problem for transforms of displacements is used to calculate a transfer function. A quadratic programming problem with equality and inequality restrictions is obtained by approximating the original contact problem. To solve this problem numerically an algorithm based on the conjugate gradient method is used. Some regularities of contact interaction have been established.