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Том 54, № 5 (2019)

Article

Stabilization Algorithm for Periodic Affine Systems

Slyn’ko V.

Аннотация

In this paper, we propose a stabilization algorithm for affine periodic systems. Linear feedback with a variable matrix that is a periodic and piecewise-constant function of time is synthesized. An example of damping resonant parametric oscillations of a linear oscillator is provided.

Mechanics of Solids. 2019;54(5):623-631
pages 623-631 views

On Motion of Chaplygin Sleigh on a Horizontal Plane with Dry Friction

Karapetyan A., Shamin A.

Аннотация

This paper considers the motion problem of the Chaplygin sleigh, a rigid body resting on a horizontal plane and equipped with three legs, one of which ends with a semicircular blade orthogonal to the supporting plane. As in the Chaplygin problem, it is assumed that the blade cannot move perpendicularly to its plane but can slide without friction along the straight intersection between the blade plane and the reference plane and rotate around its vertical radius. In contrast to the Chaplygin problem, it is assumed that at other support points the body is affected by dry friction forces. The equations of motion of the system are derived and the body motion is qualitatively analyzed.

Mechanics of Solids. 2019;54(5):632-637
pages 632-637 views

On Rolling of a Heavy Disk on a Surface of Revolution with Negative Curvature

Sumbatov A.

Аннотация

In the problem on rolling of a heavy round homogeneous disk on a surface of revolution with a negative Gaussian curvature, the classical nonholonomic model is used in which, at each moment, the instantaneous velocity of the current drive point of the disk touching the support is zero. Stationary motions of the disk are found. We note that, within the nonholonomic model the tangential component of the reacton for a stationary motion can be larger than the pressure force. This means that such motion in practice cannot be implemented or observed if we assume that the force that provides the no-slip condition is the dry friction force with a coefficient between zero and unity. For stationary motions of the disk the conditions of the stability in the first approximation are obtained. The results of the numerical simulation of the rolling motion of the disk without slip while the mechanical energy dissipation occurs, are presented. The purpose of these studies was to verify the adequacy of the assumed nonholonomic model of coin movements observed in practice in the entertaining coinboxes of the plastic funnel type.

Mechanics of Solids. 2019;54(5):638-651
pages 638-651 views

Resonant Motions of the Statically Stable Lagrange Spinning Top

Zabolotnov Y.

Аннотация

In this paper, we analyze lower-order resonances during the motion of the Lagrange spinning top with a small mass asymmetry. The conditions for the implementation of long resonance modes obtained by both the averaging method in the linear case and the method of integral manifolds for considerable nutation angles are compared. The motion of a heavy rigid body with an elongated inertia ellipsoid is considered in the vicinity of the lower statically stable equilibrium position. We present a theorem that justifies the application of the method of integral manifolds for considerable nutation angles when a rigid body moves around a fixed point. The resonance capture conditions in the nonlinear case are estimated for nutation angles not exceeding ρ/2. Numerical examples illustrating the effect of nonlinearities on the resonant motion of the Lagrange spinning top are considered.

Mechanics of Solids. 2019;54(5):652-668
pages 652-668 views

Decomposition Synthesis of the Control System of Electromechanical Objects in Conditions of Incomplete Information

Antipov A., Krasnov D., Utkin A.

Аннотация

The problem of the synthesis of a tracking system for electromechanical control objects is considered when measuring only the generalized coordinates of the mechanical subsystem. It is assumed that external and parametric perturbations act on the system. Assuming the smoothness of external perturbations and uncertainties, the mathematical model of the control object is presented in the form of a joint block form of controllability and observability relative to the coordinate basis of mixed variables (functions of state variables, external influences, and their derivatives). Based on these conditions of the joint block form of parametric uncertainty of the matrix before the control actions, a procedure for decomposition synthesis of the basic law of discontinuous control has been developed. The basic control law ensures exponential convergence of the generalized coordinates to the reference signals. To obtain estimates of the mixed variables used in the feedback, a method for synthesizing an observer of a lower order state with large coefficients is proposed, in which the principle of complete decomposition is implemented. The results of numerical modeling of the developed algorithms for a three-link manipulator operating in a cylindrical workspace under incomplete information conditions are presented.

Mechanics of Solids. 2019;54(5):669-682
pages 669-682 views

Movement of a Heavy Rigid Body Suspended on a Cable of Variable Length with Oscillation Elimination

Russkikh S., Shklyarchuk F.

Аннотация

A nonlinear problem of terminal movement of a heavy rigid body suspended on an inextensible inertialess cable of variable length with controlled horizontal movement of the suspension point is considered. It is required to move the body for a certain time from an initial resting position to a given final resting position with oscillation elimination in the end of the operation. The law of variation in the cable length is assumed to be given, and the controlled movement of its suspension point is unknown. An approximate solution to the problem of kinematic control of the oscillations of a system described by two nonlinear differential equations with variable coefficients for moderately large angles of rotation of the strained cable and the body is sought in the form of series with unknown coefficients by the Bubnov-Galerkin method using given basic functions of time that satisfy some initial and final conditions. The acceleration of the suspension point of the cable is sought in the form of a finite series of sines with unknown coefficients. A coupled system of nonlinear algebraic equations for all unknown coefficients is obtained, which includes the equations of the Bubnov-Galerkin method and initial and final data that are not fixed when choosing the basis functions. This system of equations is solved by the method of successive approximations using, in the first approximation, solutions of the linearized equations. In the examples of a system with a cable of constant and variable length, calculations are carried out with an analysis of the accuracy of the solutions by comparing them with the numerical solutions of the nonlinear differential equations of the direct problem by the Adams method with the control laws found.

Mechanics of Solids. 2019;54(5):683-693
pages 683-693 views

Strange Behavior of Natural Oscillations of an Elastic Body with a Blunted Peak

Nazarov S.

Аннотация

The point of a peak on the surface of an elastic body Ω generates a continuous spectrum inducing wave processes in a finite volume (“black holes” for elastic waves). The spectrum of a body Ωh with a blunted peak is discrete, but the normal eigenvalues take on “strange behavior” as the length h of the broken tip tends to zero. In different situations, eigenvalues are revealed that do not leave the small neighborhood of the fixed point or, conversely, fall off along the real axis with high velocity, but smoothly decrease to the lower limit of the continuous spectrum of the body Ω. The chaotic wandering of eigenvalues above the second limit may occur. A new way of forming the continuous spectrum of the body Ω with a peak from the family of discrete spectra of the bodies Ωh with a blunted peak, h > 0, has been discovered.

Mechanics of Solids. 2019;54(5):694-708
pages 694-708 views

Contact Problem for Inhomogeneous Cylinders with Variable Poisson’s Ratio

Pozharskii D.

Аннотация

In cylindrical coordinates, the system of two elastic-equilibrium differential equations is studied under the assumption of axial symmetry and the assumption that the Poisson’s ratio is an arbitrary, sufficiently smooth, function of the radial coordinate and the modulus of rigidity is constant. It turns out that the elastic coefficient is variable with respect to the radial coordinate in this case. We propose a general representation of the solution of this system, leading to the vector Laplace equation and scalar Poisson equation such that its right-hand side depends on the Poisson’s ratio. Being projected, the vector Laplace equation is reduced to two differential equations such that one of them is the scalar Laplace equation. Using the Fourier integral transformation, we construct exact general solutions of the Laplace and Poisson equations in quadratures. We obtain the integral equation of the axially symmetric contact problem on the interaction of a rigid band with an inhomogeneous cylinder and find its regular and singular asymptotic solutions by means of the Aleksandrov method.

Mechanics of Solids. 2019;54(5):709-716
pages 709-716 views

Contact Problem for a Cylindrical Waveguide With a Periodic Structure

Chebakov M., Kolosova E.

Аннотация

The axially symmetric problem of a stamp excitation of torsional oscillations in a cylindrical waveguide with periodically changing mechanical properties along the longitudinal coordinate has been studied. A section of waveguide, corresponding to the minimal period of change in mechanical properties, can consist of any number of homogeneous regions (finite cylinders) with a different length and with various elastic constants. The method, related to the construction of the special “transition operator”, which allows finding the values of the displacement vector and stress tensors on one cross-section of a waveguide by their values on another. The distance between waveguide sections equals the value of the minimal period of changing properties of the waveguide. Relations for calculating the eigenvalues of the transition operator are obtained. The study of these eigenvalues allows defining the range of frequencies when both damping and continuous oscillations can propagate in the waveguide. Contact strains under a stamp are determined for relatively large radii of the cylinder.

Mechanics of Solids. 2019;54(5):717-725
pages 717-725 views

Cauchy Problem for the Torsional Vibration Equation of a Nonlinear-Elastic Rod of Infinite Length

Umarov K.

Аннотация

for the differential equation of torsional vibrations of an infinite nonlinear-elastic rod, the solvability of the Cauchy problem in the space of continuous functions on the real axis is studied. An explicit form of the solution of the corresponding linear partial differential equation is obtained. The time interval for the existence of the classical solution to the Cauchy problem for a nonlinear equation is found and an estimate of this local solution is obtained. Conditions for the existence of a global solution and blow-up of the solution on a finite interval are considered.

Mechanics of Solids. 2019;54(5):726-740
pages 726-740 views

A Method for Solving Problems of the Isotropic Elasticity Theory with Bulk Forces in Polynomial Representation

Kuz’menko V., Kuz’menko N., Levina L., Pen’kov V.

Аннотация

A method for solving problems in isotropic elasticity theory with polynomial bulk forces is substantiated. The existence of a basis for the state space generated by monomials of random orders, which are the components of a volumetric force, makes it possible to obtain a rigorous description of a corresponding stress-strain state for any polynomial force. Solutions of the basic mixed equilibrium problem are obtained: (1) for a truncated cylinder clamped at the base and exposed to the action of a nonconservative volume force and (2) for a heavy hemisphere clamped at the equatorial section and having a nonhomogeneous shear modulus typical for bodies with subsurface hardening.

Mechanics of Solids. 2019;54(5):741-749
pages 741-749 views

Deflection of a Thin Rectangular Plate with Free Edges under Concentrated Loads

Fedoseyev V., Yagnyatinskiy D.

Аннотация

The analytical solution for the deflection of a thin rectangular plate with free edges under concentrated loads from actuators resting on an infinitely rigid foundation is presented. The solution is based on the application of the integral transforms and takes into consideration the kinematic pairs between the plate and actuators, which are deformed according to Hooke’s law.

Mechanics of Solids. 2019;54(5):750-755
pages 750-755 views

Modeling the Dynamic Bending of Rigid-Plastic Hybrid Composite Curvilinear Plates with a Rigid Inclusion

Romanova T.

Аннотация

A general method has been developed for calculating the dynamic behavior of rigid-plastic composite layered fibrous plates with a rigid inclusion and with the hinged or clamped arbitrary smooth non-concave curvilinear contour subject to a uniformly distributed short dynamic explosive loading of high-intensity. The distribution of layers is symmetric with respect to the middle surface, and in each layer there is a family of reinforcement curvilinear fibers in the directions parallel and normal to the plate contour. The structural model of the reinforcement layer with a one-dimensional stress state in the fibers is used. Depending on the loading amplitude, different types of plate deformation are possible. Based on the principle of virtual power in combination with the d’Alembert principle, the equations of dynamic deformation are derived and their implementation conditions analyzed. The analytical expressions for assessing the limiting loads, deformation time, and residual deflections of the plates are obtained. It is shown that the variation in the reinforcement parameters significantly affects both the loading capacity of such plates and the residual deflections. Examples of numerical solutions are provided.

Mechanics of Solids. 2019;54(5):756-772
pages 756-772 views

Solving the Problem of the Bending of a Plate with Fixed Edges by Reduction to an Infinite Systems of Equations

Ivanova O., Pavlov N., Fedorov F.

Аннотация

We consider the known problem of the bending of a plate with fixed edges and a homogeneously distributed load. The solution can be found by solving infinite systems of equations. Based on the construction of a solution of biharmonic equations with boundary conditions, it is proven that these infinite systems have a finite unique solution. We use the existence of a special particular (“strictly particular”) solution (SPS) of the systems to which the solution obtained by the simple reduction method converges. Such a solution always exists, if the general system is consistent and it has the following particular properties: 1) it is a unique particular solution, which is expressed by Cramer’s formula; 2) it does not contain the nontrivial solution of the corresponding homogeneous system as an additive term; 3) the well-known principal solution of the infinite system coincides with the SPS. The SPS allows calculating the values of the plate deflection, bending moments and pressures at its contour. It is shown that the construction of the SPS (in fact, the exact solution) of the obtained infinite system does not depend on its regularity or irregularity.

Mechanics of Solids. 2019;54(5):773-779
pages 773-779 views

Destruction of the Lava Dome of an Underwater Volcano

Andryushchenko V., Goloveshkin V., Murashkin I., Kholin N.

Аннотация

The stress state and destruction of the lava dome of the underwater volcano exposed to temperature gradients and external pressure from hot magma melt and the cold water layer are investigated. Methods of the linear elasticity theory and statistical strength theory are used for mathematical modeling of the behavior of lava domes. It has been revealed in the course of evaluation calculations that the role of temperature gradients in the lava dome destruction process is as important as that of an external pressure. It is established that the lava dome is destroyed layer by layer, and the time of this destruction is determined in this paper.

Mechanics of Solids. 2019;54(5):780-785
pages 780-785 views

A Model for Refined Calculation of the Stress-Strain State of Sandwich Conical Irregular Shells

Bakulin V.

Аннотация

Based on layer-by-layer analysis, we consider the construction of a model of two types of finite elements (FE) with natural curvature: a two-dimensional FE model of the moment of load-bearing layers and the three-dimensional FE of a filler for a more refined study of the stress-strain state (SSS) in the layers using the example of three-layer conical, generally irregular shells. The algorithm for constructing the model can avoid the errors caused by a discontinuity in the generalized displacements on the docking surfaces of the FEs of the bearing layers and filler. The model can more adequately to take into account the features of layered-heterogeneous structure (for example, a continuity violation of one or more layers), the moment state of bearing layers, the three-dimensional SSS in the filler, as well as different conditions for fixing and loading layers. In this case, it is possible to calculate the sandwich shells with variable geometric and physicomechanical properties and take into account the change in these properties and parameters of the SSS not only by meridional and circumferential coordinates, but also by the thickness of the shell and filler layer. As an example, the problem of determining the SSS parameters of a sandwich conical shell with blind cutouts in the inner bearing layer is solved, while the outer bearing layer and the filler layer remain continuous.

Mechanics of Solids. 2019;54(5):786-796
pages 786-796 views

Nonlinear Model of Deformation of Crystalline Media Allowing for Martensitic Transformations: Plane Deformation

Aero E., Bulygin A., Pavlov Y.

Аннотация

This article is devoted to development of mathematical solutions of statics equations of plane nonlinear deformation of crystalline media with a complex lattice allowing for martensitic transformations. Statics equations comprised of a set of four coupled nonlinear equations are reduced to a set of separate equations. The macrodisplacement vector is sought in the Papkovich-Neuber form. The microdisplacement vector is determined by the sine-Gordon equation with a variable coefficient (amplitude) before the sine and Poisson’s equation. For the case of constant amplitude the class of doubly periodic solutions has been determined which are expressed via elliptical Jacobian functions. It has been demonstrated that nonlinear theory leads to a combination of solutions describing fragmentation of the crystalline medium, occurrence of structural imperfections of various types, phase transformations, and other peculiarities of deformation which occur under the action of intensive loads and are not described by classical continuum mechanics.

Mechanics of Solids. 2019;54(5):797-806
pages 797-806 views

Analytical Solution to the Boundary Value Problem of Steady Creep of a Nonaxisymmetric Thick-Walled Tube under the Action of Internal Pressure

Moskalik A., Radchenko V.

Аннотация

The boundary value problem of the steady-state creep of a nonaxisymmetric thick-walled tube under the action of internal pressure is considered under the assumption that the material is incompressible for creep deformations. An approximate analytical solution of the problem by the small parameter method up to the third order of approximation, inclusively, for a plane deformed state is presented. The small parameter in the problem is the displacement of the centers of the inner and outer radii of the tube. The error in solving the problem is estimated by comparing the approximate analytical solution with a numerical solution obtained by the finite element method for some special cases. The analytical and numerical solutions for stresses are analyzed by dependence on the small parameter and the stress exponent of the steady-state creep. The error of the approximate analytical solution for the stress tensor components in the minimum cross section is analyzed in accordance with the industrial standards for the permissible deviation of the wall thickness in the industrial production of tubes.

Mechanics of Solids. 2019;54(5):807-818
pages 807-818 views

Murnaghan’s Elastoplastic Material Model

Shved O.

Аннотация

Murnaghan’s model of elastic material is generalized to an elastoplastic material. It is assumed that the active process occurs via alternating between the plastic and elastic states. The deformation gradient is replaced by a nonsingular tensor, and constitutive equations in the finite form are written. In addition to Green’s postulate concerning the existence of a stress potential, it is assumed that there exists a stress rate potential, for which rate the objective derivative is uniquely determined. This derivative can be obtained via modifying Green-Nakhdi derivative, wherein the spin of the rotation tensor accompanying the general deformation is replaced by a spin of the rotation tensor accompanying the elastic deformation. A deviator section of the yield surface in the stress space is determined, and differential constitutive equations are formulated. A decrease in the elastic deformation anisotropy under flowing that can lead to an onset of macrocracks is described.

Mechanics of Solids. 2019;54(5):819-831
pages 819-831 views

Modeling the Elasto-Visco-Plastic Bending of Spatially Reinforced Plates Accounting for the Strain-Rate Sensitivity of Composition Components

Yankovskii A.

Аннотация

A version of the model of elasto-visco-plastic deformation is developed taking into account the strain-rate sensitivity. This model leads, as a special case, to the equations of the Prandtl-Reuss-Hill flow theory resolved for stress rates. This mechanical model is used in deriving the structural relations describing inelastic deformation of spatially reinforced composite media based on an algorithm of time steps. In addition, a mathematical model is constructed for elasto-visco-plastic bending behavior of reinforced plates when the possible weak resistance to transverse shear is taken into account in the framework of the traditional nonclassical Hambardzumyan theory and the geometric nonlinearity of the problem is given in the Karman approximation. The solution to the initial boundary value problem is based on an explicit numerical cross-type scheme. The bending inelastic dynamic deformation is investigated for plane and spatially reinforced fiberglass and metal composite rectangular plates under the action of a load caused by an air blast wave. It is shown that the neglect of the strain-rate sensitivity of the composition components most commonly leads to an overestimation of the calculated deflections and the strain state characteristics of these components. It is demonstrated that for relatively thick structures the replacement of the flat reinforcement structure by the spatial structure implies a decrease in the intensity of the binder deformation by tens of percent as well as to a decrease in the plate deflections (it is insignificant in the case of metal composite structures and is on the order of tens of percent in the case of fiberglass plates). For relatively thin structures the replacement of the plane reinforcement structure by the spatial reinforcement structure does not result in a decrease in their flexibility in the transverse direction and in a decrease of the strain state characteristics of the composition components.

Mechanics of Solids. 2019;54(5):832-852
pages 832-852 views

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