Vol 53, No 2S (2018): Suppl

In a homogeneous gravitational field, the movement of a rigid body around a fixed point is considered. No dynamic symmetry of the body is assumed, and its centroid lies on the perpendicular to a circular section of the moment ellipsoid, restored from the fixed point. Within this mass geometry a regular axial precession of the body is possible, such that the precession axis does not coincide with the vertical axis (the Grioli precession). We investigate the stability of that precession for the special case where the ellipsoid is strongly stretched along an axis close to the axis containing the centroid. The regular precession of the body over an axis inclined with respect to vertical axis has been discovered by Grioli in 1947 (see [1]). This investigation is extended in [2–5]: in particular, it is shown that no heavy rigid body has regular precessions different from the classical precession of a dynamically symmetric body (in the Lagrange case) and the regular precession described by Grioli. The history of the discovery and investigation of the precession motions of rigid bodies is presented in [6–8]. In [9], the investigation of the stability problem for the Grioli precession is originated. Further, various ways that the problem has been posed (both numerical and analytical) are considered in [10–15]. However, no complete or strict solution for all admissible values of parameters of the problem has yet been obtained. Below, we present new results on the orbital stability of the Grioli precession.

This paper considers the stone towers that have recently attracted attention at that, at first glance, seem impossible. It is shown that a body with three points of support (tripod) on an uneven surface, such as, in particular, an inclined plane, can serve as the base model of such a system. The necessary and sufficient equilibrium conditions are obtained of such towers. Estimates of the coefficient of friction for the supports that would ensure the equilibrium in case of ground vibration and wind are determined.

The mechanisms obtained from pendulum systems with a single pivot point by adding a holonomic restraint connecting the kinematic chain to the second pivot point have been considered. The features of the configuration space that affect the dynamics of such objects have been discussed. Sufficient conditions for global controllability in the entire phase space have been formulated and an example of a mechanism (with configurations on a sphere with two handles) that is globally controlled under the action of a single limited control force has been given.

The movement of the simplest model of a statically unbalanced rotor under the condition of the action of a constant external torque is considered, taking into account the forces of external and internal damping. Two modes of stationary motion of the rotor are investigated: synchronous whirling and asynchronous self-excited oscillations. Using the system of equations that describes the rotor dynamics in polar coordinates, conditions for the existence and stability of both types of stationary modes are obtained.

Linear stationary systems with single-control (perturbing) action are considered. The impulse of control actions is considered limited. Some properties of the boundaries of attainability domains are studied. It is shown that the boundary of the attainability domain can have flat regions, regions of ruled surfaces, edges, and conical angular points. An attainability domain is not strictly convex if there are straight edges and/or flat regions on the boundary. The behavior of the boundaries of the attainability domains with increasing time is studied. A third-order system with a threefold zero eigenvalue (triple integrator) is considered as an example. The structure of the attainability domain of this system is analytically investigated in three-dimensional space. An attainability domain is constructed numerically for some time values.

The conditions for the existence of a linear invariant system of Poincaré–Zhukovsky equations have been found. In a linear invariant system, three configuration conditions have been obtained that are sufficient to allow a mechanical system without dynamic symmetry to undergo regular precession. The explicit expression of the moments of inertia of a system consisting of a rigid body with an ellipsoidal cavity filled an ideal vorticity fluid is given in terms of the cavity dimensions; the velocities of precession and self-rotation are found. The particular case of the permanent rotation of an asymmetric rigid shell around the angular momentum vector is considered; in this case, any axis rigidly bound to the shell can be used as the axis of permanent rotation.

A method for solving self-adjoint eigenproblems for linear Hamiltonian systems with equation, coefficient, and boundary conditions nonlinearly dependent on the spectral parameter is presented. The suggested approach is based on the iterative Newton procedure with spectral correction. The fast convergence of the method is demonstrated, and two-sided estimates of the eigenvalue sought are obtained. The results of the test application of the outlined algorithm are presented for the problem of the transverse natural oscillations of nonhomogeneous rods with a density defect, using the Euler–Bernoulli, Rayleigh, and Timoshenko models.

Second-order partial differential equations are considered that describe the behavior of elastic bodies and have singular solutions. In contrast to the common differential calculus based on the analysis of the function behavior in the neighborhood of a point at infinitesimal variation of arguments, the nonlocal function and its derivative, which describe the behavior of the function in a small, but finite, interval of variation of the argument, are introduced. As a result, the order of the equations under consideration increases up to fifth, and the solution to the traditionally singular problems of mathematical physics appears regular. The solution to the nonlocal problem depends on the constant coefficient, which is suggested to be determined experimentally. Possible applications we consider include the generalized solutions to the equation of mathematical physics and mechanics in Cartesian, polar, and spherical coordinates, governing the bending of the thin membrane and the stress state of the elastic orthotropic sphere.