Vol 10, No 3 (2018)

Two approaches to 3D path coordinates used when solving a path following problem for aerial vehicles are proposed. The first approach lies in reducing the problem to a two-dimensional case by projection. The second approach is based on introducing an adaptive frame at the target point. The choice of the adaptive frame determines the complexity of the algorithm of the control synthesis. It is shown that the parallel transport frame, or the Bishop frame, is most convenient.

A two-level mathematical model is constructed to describe the thermomechanical interaction between structural elements of a composite (nanoclusters formed by randomly distributed anisotropic single-walled carbon nanotubes and matrix particles) and an isotropic medium possessing the desired thermoelastic characteristics. This model was first employed to obtain the thermoelastic properties of a nanocluster by the self-consistency method and then the same technique was used to describe the thermomechanical interaction of nanoclusters with an isotropic matrix of the composite. A comparative analysis of the calculated dependences for the elastic moduli of the composite and its thermal coefficient of linear expansion was carried out with two-sided estimates of these characteristics based on the dual variational formulation of the thermoelasticity problem. For comparison, the results of a numerical experiment are also used. The presented relationships make it possible to predict the thermoelastic properties of promising composites reinforced by nanoclusters.

A mathematical model describing the propagation of Lamb waves in layered anisotropic media is introduced. The model is based on the six-dimensional complex Cauchy formalism, which makes it possible to derive the dispersion equation for Lamb waves in layered media with an arbitrary elastic anisotropy. Problems of its numerical realization are considered.

An investigation of a self-similar solution of a coupled problem on the creeping flows of a viscous fluid in a hydraulic fracture and the strain and flow in the external poroelastic medium induced by them. The process is governed by injection fluid into a well. The flow in the fracture is described by the Stokes hydrodynamic equations in the approximation of the lubricant layer. The external problem is described by the equations of poroelasticity. The variant of a uniform pressure in the fracture is considered for three-dimensional and two-dimensional cases. In the second case, a self-similar solution can be obtained in an analytical form.

We investigate the precision in the readings of the star tracker BOKZ-M60 obtained in the dynamic field testbed of the Space Research Institute of the Russian Academy of Sciences (SRI RAS). The testbed features a mobile platform, on which a star tracker is mounted, which scans the starry sky. The platform movement is controlled by a PC, however, the values of the rotation angles in the platform hinges are not known with the desired degree of accuracy. In testing star trackers, the error in the known platform orientation must be around 1″, which corresponds to an error in determining the linear coordinates of the platform reference points of less than 3 μm. It is difficult and expensive to organize outside measurements of the movement of a mobile object with such accuracy. For this reason, the following accuracy estimation scheme is used. We consider comparatively long series of the determined orientation parameters obtained by the tracker in the testbed. With the help of a suitable mathematical model, a detailed reconstruction of the actual rotational motion of the tracker is performed based on these measurements. In this reconstruction, the residuals in the tracker readings are calculated. Based on the analysis of the residuals, conclusions are drawn about the accuracy of the tracker. As it turns out, the accuracy of determining the optical axis of the tracker lens depends on the angular velocity of the platform and lies in the range from 2″ to 25″

A numerical stabilization problem by the boundary conditions of the given initial flow is considered for a two-dimensional system of Navier-Stokes equations that approximately describes the motion of a viscous incompressible fluid in a rectangular cell under an external electro-magnetic force and has an unstable quasi-stationary solution with a four-vortex structure for the selected values of the parameters. A mathematical statement of the problem, a method to solve it, and the results of the numerical computations are presented.