Vol 87, No 5 (2023)

The nonlinear problem of the Watt regulator dynamics is investigated. It is assumed to be installed on a machine that performs the specific harmonic oscillations of small amplitude along the vertical. Viscous friction forces is believed to arise in the hinges of the regulator, and these forces are small. In the main operating mode of the regulator, its rods, carrying massive weights, are deflected from the downward vertical by a constant acute angle. If friction and vertical oscillations of the machine are neglected, then we obtain an approximate problem in which the dynamics of the regulator is described by an autonomous Hamiltonian system with one degree of freedom. On the phase portrait of the approximate problem, the operating mode corresponds to a singular point of the center type. The trajectories surrounding this point lie inside the separatrix, which is a homoclinic doubly asymptotic trajectory that passes through the equilibrium position corresponding to the vertical position of the rods with weights. In the phase portrait, this position corresponds to a saddle singular point. The Melnikov method is used to obtain the splitting condition for the unperturbed separatrix in the complete perturbed problem, taking into account dissipation in the hinges and vertical vibrations of the machine.

An analytical solution of the problem of the movement of a spacecraft from the starting point to the final point in a certain time is given. First, the method of fast sine expansions is used. The space problem considered here is essentially non-linear, what necessitates the use of trigonometric interpolation methods, which surpass all known interpolations in accuracy and simplicity. In this case, the problem of calculating Fourier coefficients by integral formulas is replaced by the solution of an orthogonal interpolation system. In this regard, two cases are considered on the segment \(\left[ {0,a} \right]\): universal interpolation and trigonometric sine and cosine interpolations. A theorem on the rapid decrease of expansion coefficients is proved, and a compact formula for calculating the interpolation coefficients is obtained. A general theory of fast expansions is given. It is shown that in this case, the Fourier coefficients decrease significantly faster with the growth of the ordinal number compared to the Fourier coefficients in the classical case. This property makes it possible to significantly reduce the number of terms taken into account in the Fourier series, significantly increase the accuracy of calculations and reduce the amount of calculations on a computer. The analysis of the obtained solutions of the spacecraft motion problem is carried out and their comparison with the exact solution of the test problem is proposed. An approximate solution by the method of fast expansions can be taken as an exact one, since the input data of the problem used from reference books have a higher error.

An oscillatory system with an excitation mechanism as in a Rayleigh oscillator, but with a nonlinear (cubic) returning force, is investigated. Using the accelerated convergence method and the continuation procedure for the parameter, limit cycles are constructed and the amplitudes and periods of self-oscillations are calculated. This is done for a wide range of feedback coefficient values, in which this coefficient is not asymptotically small or large. The proposed iterative procedure allows to achieve the specified accuracy of calculations. The analysis of the features of the limit cycle caused by an increase in the self-excitation coefficient is carried out. The results obtained are compared with the self-oscillations of a classical Rayleigh oscillator with a linear returning force.

The motion of a heavy rigid body with a fixed point in a uniform gravitational field is considered. It is assumed that the main moments of inertia of the body for the fixed point satisfy the condition of D.N. Goryachev–S.A. Chaplygin, i.e., they are in the ratio 1 : 4 : 1. In contrast to the integrable case of D.N. Goryachev–S.A. Chaplygin, no additional restrictions are imposed on the position of the center of mass of the body. The problem of orbital stability of pendulum periodic motions of the body is investigated. In the neighborhood of periodic motions, local variables are introduced and equations of perturbed motion are obtained. On the basis of a linear analysis of stability, the orbital instability of pendulum rotations for all values of the parameters has been concluded. It has been established that, depending on the values of the parameters, pendulum oscillations can be both orbitally unstable and orbitally stable in a linear approximation. For pendulum oscillations that are stable in the linear approximation, based on the methods of KAM theory, a nonlinear analysis is performed and rigorous conclusions about the orbital stability are obtained.

The paper considers the problem of damping vibrations of a membrane and a plate with the help of forces distributed over their entire area. The proposed method allows us to consider restrictions not only on the absolute value of the control, but also on the absolute value of the derivatives of the functions that specify the control. Sufficient conditions are given for the initial conditions under which the problem of bringing the system to rest in a finite time is solvable, and the time of bringing to rest is estimated.

New mathematical models of a sessile drop and a captive bubble are constructed taking into account the size dependence of surface tension. If the Tolman length tends to zero the well-known Bashforth–Adams model can be considered as a special case of the constructed models. Numerical calculations of the contact angles are carried out for various numeric values of the equilibrium volume. The study shows that the size dependence of the surface tension leads to a violation of the consistency condition between the contact angles of a drop and a bubble in an external force field.

The paper presents closed-form analytical solution to the plane-strain problem of stress relaxation in a bended plate with tension-compression asymmetry (TCA) in viscous properties. Reversible and irreversible strains are assumed to be finite. We utilize a linear viscous model with equivalent stress that is piecewise linear function of the principal stresses with TCA parameter. The specific features of the solution are discussed.