Том 53, № 1S (2018): Suppl

The problem of control of a plane three-link inverted pendulum by means of one or two torques applied at ball joints is considered. This pendulum is an example of a nonlinear underactuated mechanical system, i.e., a system in which the number of degrees of freedom exceeds the dimension of the vector of the generalized controlling force. A three-link pendulum has eight different equilibrium positions, at which some links are directed upwards, and other are directed downwards. All equilibrium positions, except the lower position, are unstable. The pendulum controllability is examined in the linear approximation in the neighborhood of these equilibrium positions for different options of control: by means of external torques applied to some links, or by means of internal torques at ball joints. In the cases of controllability, the control limited by module is constructed in the feedback form to move the pendulum from the neighborhood of the given equilibrium position to the equilibrium position over a finite time period.

We investigate the optimal control problem for the reorientation of a perfectly rigid circular top. For the criterion of the maneuver’s efficiency, we take the integral-square functional characterizing the total energy consumption. The control is the principal momentum of the applied external forces. In this problem, we obtain (for the first time) a scalar nonlinear differential equation such that its solution characterizes the second power of the absolute value of the extremal angular velocity vector. For polynomials depending on time, the family of solutions of the equation is considered. Numerical examples are provided.

This paper focuses on a spheroid with the center of mass coinciding with the geometric center moving in a horizontal plane with friction. For a certain class of initial conditions, a qualitative analysis of the dynamics is given depending on the presence of sliding, rolling or pivoting friction. A geometric interpretation of the results is given. The dynamics of a rigid rotation body on a plane with friction was studied within the framework of classical models of friction (dry or viscous, as well as a model of an absolutely rough plane), for example, see [1, 2]. A global qualitative analysis of the dynamics was carried out within the framework of interrelated friction models [3–5] for a uniform ball [6, 7] as well as a ball [8] and an ellipsoid [9] with a displaced center of mass. The stability of stationary motions in the case of a uniform spheroid on a plane with viscous friction is discussed in [10].

Sandwich shells with transversely soft cores and skin layers supported along the edges by a thin deformable diaphragm with a ruled middle surface are considered. Delamination zones are assumed on the contact surfaces between the core and skin layers. The refined geometrically nonlinear theory has been constructed for these type of structures at small deformations and moderate displacements, which can describe prebuckling deformation and reveal all possible buckling mode shapes of skin layers (antisymmetric, symmetric, mixed bending and mixed bending-shear, as well as arbitrary mode shapes comprising all listed modes) and of the reinforcing diaphragm. This theory is based on the introduction of interaction forces between the core and skin layers and between the core and reinforcing diaphragm at each point of the contact surfaces as unknowns. The previously proposed variant of the generalized Lagrangian variational principle has been utilized in the derivation of the governing equations, static boundary conditions for the shell and reinforcing diaphragm, and kinematic coupling conditions between the core and skin and between the core and support diaphragm.

A one-dimensional model of harmonic oscillations of a junction of several thin elastic rods has been developed. In contrast to the classical model of a single rod, the constructed model of the junction is not purely differential but includes new algebraic unknowns and algebraic equations evoked by the so-called readily movable elements of the structure. The asymptotic representations have been found for frequencies and natural modes of the elastic body oscillations and estimates of asymptotic residues have been obtained.

An analytical solution of a periodic contact problem on the penetration of a rigid wavy surface represented by two cosine harmonics into an elastic half-plane without accounting for the frictional forces is obtained. It is shown that the presence of cosine harmonics of the second scale for moderate and large loads causes oscillations of the pressure distributions, which depend linearly on its amplitude and nonlinearly on its period. The presence of the oscillations of the contact characteristics—the nominal and maximal pressures—is discovered.

Problems of elasticity theory of inhomogeneous bodies (such as a plate) about the of plane stress state are considered in spatial formulation. It is assumed that the material is inhomogeneous by thickness and its properties change symmetrically with respect to its median plane. A method of solving such types of problems using complex potentials is elaborated. The basic correlations of the method are obtained. The exact analytical solutions of two problems are presented: about the inhomogeneous radial stretching of the contour of a circular disk and about the inhomogeneous radial stretching of the bypass of a circular aperture in an infinite plate.