Vol 53, No 5 (2018)

To obtain information about the inertial trihedron on board a moving object, it suffices to observe the trajectory of a three-dimensional isotropic oscillator with the center of suspension on this object. In the most general case, the trajectory of such an oscillator is an ellipse that does not have an angular velocity relative to the inertial space. In this paper, we construct an oscillator control stabilizing an elliptical trajectory with a nonzero quadrature and prove the stability of this control.

The article deals with the question of what is a type of flexural edge wave on a circular plate. It is shown that, in contrast to the case of a rectilinear plate, the flexural edge wave on a circular plate is a wave of fundamentally different type, namely a whispering gallery wave. With an increase in the wave number, this wave gradually turns into an analogue of the Konenkov wave, but this happens in the region of very short waves. The dependence on Poisson’s ratio (the “critical” value of the harmonic number, at which the wave transformation from whispering gallery type to the Konenkov type occurs) is constructed. The certain conditions, under which the transition region does not go beyond the scope of the Kirchhoff theory, are determined.

The statements and exact self-similar solutions of diffusion-vortex problems in terms of stresses simulating a nonstationary one-dimensional shear in some curvilinear orthogonal coordinate systems of a two-constant stiff-viscous plastic medium (Bingham body) are analyzed. Such problems include the diffusion of plane and axisymmetric vortex layers, as well as the diffusion of avortex filament. The shear occurs in regions of unlimited space expanding with time with a pre-unknown boundary, and a possible way of specifying an additional condition at infinity is described. A generalized vortex diffusion is introduced into consideration, containing a formulation with several parameters, including the order of the singularity peculiarities at zero. Self-similar solutions are constructed in which the order of the singularity corresponds to or does not correspond to the type of shift in the selected coordinate system.

The first part of the article provides an overview of the work on the differential equations of the spacecraft (SC) orbit orientation and the problem of optimal reorientation of a spacecraft orbit in an inertial coordinate system by means of reactive acceleration orthogonal to the osculating plane of the spacecraft. The theory of solving the problem of the optimal reorientation of the orbit of the spacecraft using the quaternionic differential equation for the orientation of the orbital coordinate system in a non-linear continuous formulation (using limited (small) thrust) is presented. As a minimized quality functional, a combined functional is used equal to the weighted sum of the reorientation time and thrust impulse (characteristic speed) during the reorientation of the orbit of the spacecraft (special cases of this functional are the speed response case and the characteristic speed minimization separately).

The theory outlined in the first part of the article is used in the second part of the article to build in a strict non-linear formulation of the new theory and new algorithms for numerical solution of the problem of the optimal reorientation of the spacecraft orbit in the inertial coordinate system by means of pulsed (high) thrust, orthogonal to the plane of an osculating orbit, using the quaternionic differential equation for the orientation of the orbital coordinate system for an unfixed number of pulses of reactive thrust. The constructed algorithms allow for the numerical solution of the problem to determine the optimal moments of switching on a reactive engine, the optimal values of reactive acceleration pulses and their optimal number. Examples are given of a numerical solution of the problem of optimal impulse reorientation of the orbit of the spacecraft, demonstrating the capabilities of the proposed method.

Based on the Kirchhoff-Love theory in a geometrically nonlinear formulation, the problem of vibrations of a viscoelastic reinforced anisotropic plate is considered. The problem is solved using the Bubnov-Galerkin method founded on a polynomial approximation of the deflections in combination with a numerical method based on the using quadrature formulas. An analysis of the use of rheological viscosity parameters for structures with different directions of reinforced fibers is presented. The effect of viscoelastic properties and directions of reinforced fibers on the plate oscillation process is exemplified by real materials.

A mathematical model of elastic oscillations of a rod under the influence of an external harmonic load, taking into account the relaxation properties and forces of the medium resistance, has been developed. The derivation of the differential equation of the model is based on taking into account the time dependence of the stresses and strains in the formula of Hooke’s law, which, when presented in this way, coincides with the formula of the complicated Maxwell and Kelvin-Voigt models. The study of the model using numerical method showed that when the frequency of the natural oscillations of the rod coincides with the frequency of the external load oscillations (if the resistance of the medium and its relaxation properties are not taken into account), the amplitude of the oscillations (resonance) increases unlimited in time. When taking into account the resistance and relaxation properties of the medium at resonant frequencies, the amplitude of oscillations stabilizes on a value depending on the values of the resistance and relaxation coefficients. At frequencies close to resonant, bifurcation oscillations (beats) are observed, at which there is a periodic increase and decrease of the amplitude of oscillations. At frequencies substantially different from resonant ones, in the case of taking into account resistance forces and relaxation properties of materials, bifurcation oscillations are not observed. In this case, the amplitude of oscillations is stabilized in time at a value depending on the amplitude of oscillations of the external load, the resistance coefficient and the relaxation coefficients.