


Vol 53, No 1S (2018): Suppl
- Year: 2018
- Articles: 15
- URL: https://ogarev-online.ru/0025-6544/issue/view/9938
Article
Feliks Leonidovich Chernousko (on His Eightieth Birthday)



Possibility of a Non-Reverse Periodic Rectilinear Motion of a Two-Body System on a Rough Plane
Abstract
A periodic rectilinear motion of a two-body system along a rough plane is considered. The system is controlled by the force of interaction of the bodies. A periodic motion is defined as a motion in which the distance between the bodies and their velocities relative to the plane are represented by time-periodic functions with the same period. The friction that acts between the bodies and the plane is Coulomb’s dry friction. Necessary and sufficient conditions for the possibility of a periodic non-reverse motion of the system, in which neither of the bodies changes the direction of its motion, are proved. These conditions are expressed by inequalities that involve the masses of the system’s bodies and the coefficients of friction of these bodies against the underlying plane.



The Control of a Three-Link Inverted Pendulum Near the Equilibrium Point
Abstract
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.



Nonlinear Regulators in the Position Stabilization Problem of the Holonomic Mechanical System
Abstract
The application of nonlinear proportional-integral and proportional-integro-differential regulators in the program position stabilization problem of a holonomic mechanical system is investigated. To this end, the method of Lyapunov functionals is developed in the stability problem of Volterra integro-differential equations. As an example, the global regulation problem of a three-link planar manipulator is solved.



On the Differential Equation for the Square Modulus of the Extremal Angular Velocity Vector in the Least-Energy Reorientation Problem for a Circular Top
Abstract
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.



Periodic Regime of Motion of a Vibratory Robot under a Control Constraint
Abstract
A class of motions of a vibratory robot consisting of a body with a pendulum inside it, on the angular acceleration of which a constraint is imposed, is considered. The limits of the angular acceleration correspond to the technical characteristics of the motor providing the pendulum rotation. Some phase constraints are imposed on the system and the control is proposed which satisfies them and provides periodic motions of the robot. The control during each period consists of seven intervals; choice of control at each of them is conditioned by velocity maximization. For considered control law numerical experiments are carried out and comparison with other controls is presented.



Spheroid Dynamics on a Horizontal Plane with Friction
Abstract
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].



Optimal Suppression of Vibrations of a Moving Elastic Web
Abstract
The translational movement of an elastic web (panel) performing transverse vibrations caused by initial disturbances is considered. It is supposed that the web moving with a constant translational velocity is described by the model of an elastic panel (beam) with supported edges of the examined span. The problem of the optimal suppression of vibrations of a multispan panel (web) supported at discrete points is formulated with consideration of forces applied to the web. In order to solve the optimization problem, we use modern methods developed with the control theory of distributed parameter systems described by partial differential equations.



Nonlinear Theory of Sandwich Shells with a Transversely Soft Core Containing Delamination Zones and Edge Support Diaphragm
Abstract
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.



Modeling the Long-Term Strength of Metals in an Unsteady Complex Stress State
Abstract
The known results of tests for long-term strength in an unsteady complex stress state are simulated using the kinetic theory. Experimental data are usually described using a vector damage parameter with a piecewise-constant damage accumulation rate. The long-term strength of tubular samples is simulated under the simultaneous action of a constant axial stress and a shear stress once or periodically changing sign. To describe the known effect in which the time to fracture in a uniaxial plane stress state is several times smaller than the time to fracture under a uniaxial tension, a variant of Yu.N. Rabotnov’s kinetic theory with additional consideration of the anisotropy of the material is proposed. The long-term strength with an abrupt change in the stress intensity is simulated by two methods: with allowance for the damage accumulation only in the creep process and with allowance for the additional damage accumulation under instant loading. All the variants of the kinetic equations proposed lead to a good agreement between the experimental and theoretical values of the time to fracture.



Asymptotics of Natural Oscillations of Elastic Junctions with Readily Movable Elements
Abstract
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.



The Contact Problem with the Bulk Application of Intermolecular Interaction Forces: Influence Function for an Inhomogeneous Elastic Half-Space
Abstract
In this paper, we consider a contact problem for an elastic half-space that is nonuniform over depth in the presence of applied bulk intermolecular interaction forces. Our attention is focused on constructing an influence function for this problem. A solution of the Mindlin problem on the action of a concentrated vertical force in an elastic half-space that is inhomogeneous over depth is obtained by using the perturbation method. The influence function is analyzed and some of its features are revealed.



Periodic Contact Problem for a Surface with Two-Scale Waviness
Abstract
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.



Periodic Crack Systems in an Elastic Wedge
Abstract
We investigate three-dimensional periodic problems of the theory of elasticity for a wedge in the case where an infinite system of identical cracks of normal separation (mathematical cuts) is located in the middle half-plane of a wedge along the rib (at an equal distance from the rib for equal intervals between neighboring cracks). Three types of boundary-value conditions on the wedge faces are considered: the absence of tensions, the sliding sealing, and the rigid sealing. The problems are reduced to an integrodifferential equation such that the principal part is selected from its kernel; that principal part corresponds to one crack in an elastic space. To solve it, the regular asymptotical method is used. The intensity coefficients for tensions are computed at different wedge angles.



The Application of Complex Potentials to the Problems of Plane Stress State for an Inhomogeneous Material
Abstract
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.


