No 1 (2023)

The article considers the problem of regularizing the features of the classical equations of celestial mechanics and space flight mechanics (astrodynamics), which use variables that characterize the shape and size of the instantaneous orbit (trajectory) of the moving body under study, and Euler angles that describe the orientation of the used rotating (intermediate) coordinate system or the orientation of the instantaneous orbit, or the plane of the orbit of a moving body in an inertial coordinate system. Singularity-type features (division by zero) of these classical equations are generated by Euler angles and complicate the analytical and numerical study of orbital motion problems. These singularities are effectively eliminated by using the four-dimensional Euler (Rodrigues-Hamilton) parameters and Hamiltonian rotation quaternions. In this (second) part of the work, new regular quaternion models of celestial mechanics and astrodynamics are obtained that do not have the above features and are built within the framework of a perturbed spatial limited three-body problem (for example, the Earth, the Moon (or the Sun) and a spacecraft (or an asteroid)): equations of trajectory motion written in non-holonomic or orbital or ideal coordinate systems, for the description of the rotational motion of which the Euler (Rodrigues-Hamilton) parameters and quaternions of Hamilton rotations are used. New regular quaternion equations of the perturbed spatial restricted three-body problem are also obtained, constructed using two-dimensional ideal rectangular Hansen coordinates, Euler parameters and quaternion variables, as well as using complex compositions of Hansen coordinates and Euler parameters (Cayley-Klein parameters). The advantage of the proposed orbital motion equations constructed using the Euler parameters over the equations constructed using the Euler angles is due to the well-known advantages of the quaternion kinematic equations in the Euler parameters included in the proposed equations over the kinematic equations in the Euler angles included in the classical equations.

The article considers the potential problem of a deep impulsive impact at the initial moment of time on a hydrogeophysical massif, which can occur during underground (underwater) explosions, volcanic eruptions, seismic events, etc. The impact of the pulse focus was modeled by a rounded source with a unit pressure, and the sink area was modeled by a line of zero potential. A rigorous hydromechanical solution of the problem is obtained with the establishment of an analytical relationship between the physical region of the flow and the complex potential based on the theory of the function of a complex variable, that is, the use of the method of successive conformal mappings with the determination of all necessary flow characteristics. Calculation examples are given for special cases with the construction of curvilinear orthogonal hydrodynamic grids, outlines of families of lines of equal heads and stream lines, profiles of impulse sources, as well as diagrams of velocities, heads and potential flow rates.

The article deals with the problem on determining the angular position during descent on an apparatus with low lift-drag ratio. A solution is presented using the least squares method, which reduces to determining the orientation quaternion using a system of linear algebraic equations. The proposed method is based on the algebraic properties of quaternions. A vibrating string accelerometer and a fiber optic gyroscope are used to obtain acceleration and angular velocity measurements. Data on the speed and coordinates of the device are available according to the readings of the satellite navigation equipment.

 Based on the assumption about the initial deformed shape of the cross section of the pipeline, cylindrical shell, carbon nanotube (CNT) without initial stresses, the critical pressures inside and outside these structural elements are determined. The static interaction of instabilities under the action of the above factors is studied.

 Using the fast trigonometric interpolation, the problem on stresses in a rectangular bar is solved. The obtained approximate analytical solution is compared with the exact one. During this analysis the relative error of the displacement components, the stress tensor components, the discrepancy between the Lame equilibrium equations, and the discrepancy of the boundary conditions are investigated. It has been established that when using the second-order boundary function in fast expansions and a small number of terms in the Fourier series (from two to six), the maximum relative error δmath max of the displacement components and the stress tensor components is less than one percent. With an increase in the order of the boundary function and/or the number of terms N in the Fourier series, δmath decreases rapidly. Increasing the order of the boundary function is a more effective way to reduce the calculation error of δmath max than increasing the number of terms in the Fourier series. When studying the intensity of stresses σ in a bar with different overall dimensions of a rectangular cross-section, but with the same area of all sections, it has turned out that the smallest value of σmath all cross-sections is observed for a bar with a square section.

In electrically conductive composites placed in an electric field, heat is released and nonuniform temperature fields are formed. This circumstance, in turn, induces strains and stresses in such composites. The article solves a sequence of unrelated boundary value problems: electrical conductivity in a field of direct electric current, thermal conductivity, thermoelasticity. It is shown that when an electric current flows in copper-graphite and copper-filled polymer composites, displacements, deformations, and stresses occur even in the case when the components of the composite do not have a piezoelectric effect.

The results of calculations of the effective Young's modulus of longitudinally stretched twolayered plates made of identically oriented cubic crystals are presented on the basis of analytical analysis and the numerical finite element method. Analytical dependences of effective Young's modulus on Young's moduli and Poisson's ratios of crystals in layers are presented. Combinations of pairs of crystals with a significant deviation of the effective characteristics from ones found by the rule of mixtures are determined. The dependences of the effective Young's moduli on extreme values of the Young's moduli and Poisson's ratios of crystals in layers are established. They are presented graphically, and in some cases are reflected in the form of a table.