Vol 24, No 3 (2022)

Proportional-integro-differentiating (PID) controllers are widely used in solving control problems of technical systems, including mechanical ones. For this case, most of works are limited to the study of stabilization problem for steady motions and states; such studies are based on the analysis of model equations in a linear approximation. On the other hand, one of the urgent problems of controlled-motion mechanics is the problem of using PID controllers in tracking the trajectories of multi-link robotic manipulators with semi-global or global stabilization in a non-linear formulation. Practically little studied is the problem of justifying the applicability of such controllers taking into account possible delay in the feedback structure. This paper deals with such a problem. As an application of the theory developed in this paper, the control for a motion of a six-link manipulator is obtained.

Solving nonlinear differential equations with external forces is important for understanding resonant phenomena in the physics of oscillations. The article analyzes this problem basing on example of an ordinary second-order differential equation of the pendulum type, where the nonlinearity is described by a sinusoidal term. The phase plane of such an oscillator is constructed and its periodic trajectories are studied. It is illustrated that bounded nonlinearity matters only at intermediate amplitudes. The excitation of a nonlinear oscillator is carried out using a limited two–component force; the first its component corresponds to an oscillation at the resonant frequency of a linear oscillator, and the second is a limited function with a variable frequency. It is shown that with the appropriate choice of an external force, it is possible to obtain unlimited amplification of oscillations in a pendulum-type oscillator with amplitude linearly proportional to time. Spectral composition of the external force is investigated using short-time Fourier transform. It is demonstrated that in order to maintain the resonant mode, the frequency of the external force must continuously increase. Energy estimates of the external force and oscillator fluctuations depending on time are performed. The considered example is important for understanding resonant conditions in nonlinear problems.

When solving problems of aerodynamics, researchers often need to numerically solve singularly perturbed boundary value problems. In some cases, the problem can be reduced to solving a boundary value problem for an ordinary differential equation. Then it is possible to apply various numerical methods such as the grid method, the shooting method, as well as a number of projection methods, which, in turn, can form the basis of the finite element method. The grid method requires solving a system of algebraic equations, that are often nonlinear, which leads to an increase in the calculation time and to the difficulties in convergence of the approximate solution. According to the shooting method, the solution of  boundary value problem is reduced to solving a certain set of Cauchy problems. When solving stiff Cauchy problems, implicit schemes are used as a rule, but in this case the same difficulties arise as for the grid method. The transformation of the problem to the best argument $\lambda$, calculated  tangentially along the integral curve, makes it possible to increase the efficiency of explicit numerical methods. However, in cases where the growth rate of integral curves is close to exponential, the transformation to the best argument is not efficient enough. Then the best argument is modified in such a way as to smooth out this flaw. This paper investigates the application of modified best argument to the solution of the boundary value problem of an aerodynamic flow movement in case when the gas is injected at supersonic speed into a channel of variable cross-section.

For the equations of gas dynamics in Eulerian variables, a family of twolayer time-fully conservative difference schemes (FCDS) with space-profiled time weights is investigated. Nodal schemes and a class of divergent adaptive viscosities for FCDS with spacetime profiled weights connected with variable masses of moving nodal particles of the medium are developed. Considerable attention is paid to the methods of constructing regularized flows of mass, momentum and internal energy that preserve the properties of fully conservative difference schemes of this class, to the analysis of their stability and to the possibility of their use on uneven grids. The effective preservation of the internal energy balance in this class of divergent difference schemes is ensured by the absence of constantly operating sources of difference origin that produce “computational” entropy (including entropy production on the singular features of the solution). Developed schemes may be used in modelling of hightemperature flows in temperature-disequilibrium media, for example, if it is necessary to take into account the electron-ion relaxation of temperature in a short-living plasma under conditions of intense energy input.