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编号 5 (2023)
Articles
The Problem Solution on the Propagation of a Griffith Crack Based on the Equations of a Nonlinear Model
摘要
On the basis of a nonlinear model of deformation of a crystalline medium with a complex lattice, the problem of the stationary propagation of a Griffith crack under the action of homogeneous expanding stresses is posed and solved. It is shown that the stressed and deformed states of the medium are determined both by external influences on the medium and by the gradients of the optical mode (mutual displacement of atoms). The contributions from these factors are separated. Finding the components of the stress tensor and macro-displacement vector is reduced to solving Riemann–Hilbert boundary value problems. Their exact analytical solutions are obtained.
3-14
A Dynamicly Consistent Model of Normal Reactions at Points of a Mobile Platform Contact with a Surface Taking Account of the Design of Mecanum Wheels and Multicomponent Friction
摘要
The article investigates the influence of the dependence of normal reactions on motion parameters on the dynamics of a mobile platform by taking into account the design of mecanum wheels and multicomponent friction. To describe the dependence of normal reactions on motion parameters, the theorems on the change in momentum and angular momentum written for the mecanum platform are used. The influence of normal reactions on the dynamics of the mecanum platform is estimated from the results of numerical simulation. The mecanum platform dynamics model takes into account the design of the mecanum wheels and multicomponent friction. The multicomponent friction model proposed by V.F. Zhuravlev that takes into account sliding and spinning is considered. Estimates of the maximum deviations of the normal reactions of the supports, due to the dynamics of the mecanum platform, from the values of the normal reactions calculated for the mobile platform at rest (equal to 16.7% for KUKA youBot robot) are given. Inequalities that limit the maximum values of the control moments, under which the contacting rollers of the mecanum wheels do not come off from the supporting surface are obtained. Based on the simulation results, it is shown that the normal responses are changed by 5–6% of the normal response value calculated in the case of the mecanum platform at rest, which corresponds to the obtained estimates. These changes in normal reactions can lead to a decrease in the accuracy of the movement of the mecanum platform obtained with program control.
15-26
Quaternion Methods and Regular Models of Celestial Mechanics and Space Flight Mechanics: Local Regularization of the Singularities of the Equations of the Perturbed Spatial Restricted Three-Body Problem Generated by Gravitational Forces
摘要
The problem of local regularization of differential equations of a perturbed spatial restricted three-body problem is studied: elimination of singularities (dividing by zero) generated by gravity forces of differential equations of perturbed spatial motion of a material point M, which has a negligibly small mass, in the vicinity of two gravitating points M0 and M1 by writing equations of motion in rotating coordinate systems, the use of new regular variables and the regularizing transformation of time. Various systems of regular quaternion differential equations (RQDE) for this problem are obtained. The following groups of variables act as variables in these equations: (1) four-dimensional Kustaanheimo–Stiefel variables, Keplerian energies and time t, (2) distances from the point M to the points M0 and M1, modules of the vectors of the moment of velocities of the point M with respect to the points M0 and M1, Keplerian energy, time t and Euler (Rodrigues–Hamilton) parameters characterizing the orientations of the orbital coordinate systems in the inertial coordinate system; (3) two-dimensional Levi-Civita variables describing the motion of the point M in ideal coordinate systems, Keplerian energies, time t and Euler parameters characterizing the orientations of ideal coordinate systems in the inertial coordinate system and being osculating elements (slowly changing variables) for the motion of the point M in the neighborhood gravitating point M0 or M1, respectively. To construct the RQDE, the equations of the perturbed spatial restricted three-body problem, written either in nonholonomic (azimuthally free), or in orbital, or in ideal coordinate systems, were used as initial ones; “fictitious” times τ0 and τ1 are used as new independent variables (i.e., regularizing differential transformations of the Sundmann time are used) or angular variables φ0 and φ1, which are traditionally used in the study of orbital motion as part of polar coordinates. To match the two independent variables used in the vicinity of the gravitating points M0 and M1, additional differential equations are used.
The obtained various locally regular quaternion differential equations of the perturbed spatial restricted three-body problem make it possible to develop regular analytical and numerical methods for studying the motion of a body of negligibly small mass in the vicinity of two other bodies with finite masses, and also make it possible to construct regular algorithms for the numerical integration of these equations. The equations can be effectively used to study the orbital motion of celestial and cosmic bodies and spacecraft, to predict their motion, as well as to solve problems of controlling the orbital motion of spacecraft and solving problems of inertial navigation in space.
27-57
Control of the Rotation of a Solid (Spacecraft) with a Combined Optimality Criterion Based on Quaternions
摘要
The dynamic problem of the optimal rotation of a rigid body (for example, a spacecraft) from an arbitrary initial to a designated final angular position in the presence of restrictions on the control variables is studied. Rotation time is not fixed. To optimize the rotation control program, a combined quality criterion is used, the minimized functional combines the energy costs and the duration of the maneuver in a given proportion. Based on the L. S. Pontryagin maximum principle and quaternion models of controlled motion of a solid, a solution to the problem was obtained. The reorientation optimality conditions are written in analytical form, and the properties of optimal rotation are shown. To construct an optimal rotation program, formalized equations and calculation formulas are written. Optimal control is presented in the form of a synthesis. The control law is formulated as an explicit dependence of the control variables on the phase coordinates. Analytical equations and relations for finding the optimal motion are given. Key relationships are given that determine the optimal values of the parameters of the rotation control algorithm. A constructive scheme for solving the boundary value problem of the maximum principle for arbitrary rotation conditions (initial and final positions and moments of inertia of a rigid body) is also described. For a dynamically symmetric rigid body, a closed-form solution of the reorientation problem is obtained. A numerical example and the results of mathematical modeling are presented, demonstrating the practical feasibility of the developed method for controlling the attitude of a spacecraft.
58-78
Dynamic Axisymmetric Tension of a Thin Round Ideally Rigid-Plastic Layer
摘要
We consider the stress-strain state that occurs during dynamic tension of a homogeneous round layer of an incompressible ideally rigid-plastic material that obeys the Mises–Genka criterion. The upper and lower bases are stress-free, and the radial velocity is set on the lateral boundary. The possibility of thickening or thinning of the layer is taken into account, which simulates neck formation and further development of the neck. Two characteristic tension modes are revealed. First one is associated with a rather high rate of removal of the side boundary of the layer from the center, the second one is associated with acceleration. In the second case, we have carried out an analysis using the method of asymptotic integration, which makes it possible to approximately find the parameters of the stress-strain state.
79-88
Stationary Boundary Problems of Coupled Thermoelasticity for a Half-Plane and Their Solution
摘要
Using the model of coupled thermoelasticity, the boundary value problems of the dynamics of a thermoelastic half-space are solved for plane deformation with periodic surface force and thermal effects associated with the desired boundary functions by linear algebraic relations. Green’s tensors are constructed for the stated boundary value problems, using their properties, analytical solutions of these problems are obtained. To solve them, we have used the method of incomplete separation of variables, the Fourier transform, and the properties of fundamental solutions. The presented algorithm solves the classical four boundary value problems of thermoelasticity, as well as non-classical ones with coupled thermal and force characteristics at the boundary of the half-plane.
89-97
Tensors with Constant Components in the Constitutive Equations of a Hemitropic Micropolar Solids
摘要
The present paper is devoted to elastic potentials and the constitutive equations of mechanics of anisotropic micropolar solids, the kinematics of which can be specified by two independent vector fields: a contravariant field of translational displacements and a contravariant pseudovector field of microrotations of weight +1. The quadratic stress potential is represented by three constitutive tensors of the fourth rank, two of which are pseudotensor in nature and can be assigned weights –2 and –1. Such a solid is completely specified by the 171st micropolar elastic modulus. The main attention is focused on the model of a hemitropic (half-isotropic, demitropic) micropolar elastic solid characterized by nine constitutive constants. The components of the constitutive pseudo-tensor of weight ‒1 turn out to be sensitive to mirror reflection transformations in three-dimensional space. A peculiar algebraic structure of the constitutive tensors of a hemitropic solid, more precisely, their absolute analogues obtained by multiplying by integer powers of a pseudoscalar unity, is studied. It is shown that these tensors can always be constructed from isomers (isomer) of a tensor with constant components (generally insensitive to any transformations of the coordinate system) and one additional fourth-rank tensor constructed, in turn, from the components of the metric tensor.
98-110
On the Mechanical Concept of Self-Assembly of Nanomaterials
摘要
The article presents the mechanical concept of self-assembly of nanoparticles. It is assumed that nanoparticles are deformable stamps in a plane dynamic contact problem, lying on the boundary of a multilayer deformable medium. The constant vibration in the microcosm is caused by the oscillatory mode by the energy of phonons and magnons. Earlier, in the works of the authors, the mechanical concept of self-organization of nanoparticles was presented. It is based on high-frequency resonance, which causes the formation of standing waves. They localize the available aggregates of nanoparticles on the crest of standing waves. The self-assembly of nanoparticles is based on resonance, previously predicted by Academician I. I. Vorovich and inherent only in deformable dies in contact problems on a multilayer medium. Deformable nanoparticles are modeled by fractals representing packed block elements described by the Helmholtz equation. The resonance of the deformable dies allows the capture of nanoparticles, dictated by the Coulomb forces of attraction. It is shown that the combination of two fractals generates a new fractal with a combined carrier, and in the case of multiple association, a fragment of a nanomaterial is obtained. To implement the study, for the first time it was possible to construct a high-precision approximate solution of a plane contact problem on the action of a stamp of any finite size on a multilayer base. This result is dictated by the need for an analytical construction of the theory of self-assembly of nanomaterials.
111-119
Studying the Properties of Metamaterials with a Negative Poisson’s Ratio when Punched by a Rigid Impactor
摘要
Some properties of metamaterials with a negative Poisson’s ratio (auxetics) have been studied experimentally when punched along the normal by a rigid spherical impactor. Samples of a metamaterial with a chiral structure (hexachirals honeycomb) are made of e-PLA plastic using a 3D printer. In experiments, a deviation of the direction of movement of the impactor after leaving the punched sample from the approach direction (normal to the side surface) is observed. The dependence of the impactor projection direction on the orientation of the elements of chiral symmetry of the samples is established. A FE model for calculating the penetration of a chiral structure has been developed. Numerical results are presented and their agreement with experimental data is noted.
120-130
Solution of the Dynamic Lame Problem
摘要
The well-known Lame problem, posed in 1852, involves solving the static equilibrium of a parallelepiped with free side surfaces subjected to action of opposite end forces. In this article, the same problem for a more complicated case of impacts of end forces is considered.
An exact analytical solution of this problem is found.
Emphasizing the particular difficulty of solving this problem, Lamé, in his book “Leçons sur la thorie mathematique de Ielasticite des corps solides” (Paris, 1852), wrote: “C’est une sorte d’engine aussi digne d’exercer la sagasite des analystes que le fameux problem des trios corps de la Mécanique celeste”,—“This is a kind of drive, as worthy of training the clairvoyance of analysts as the famous three-body problem of celestial mechanics.” At that time, this problem was the subject of a prize from the Paris Academy of Sciences, that was intended for the one who solved the Lamé problem. Despite this, to date, no solution has been found even for a static case of this problem, not to mention the complicated version of the problem.
131-137
Fundamental Solutions of the Equations of the Oscillation Theory for Anisotropic Elastic Media
摘要
The construction of fundamental solutions in R3 for the equations of harmonic vibrations in the theory of elasticity of anisotropic elastic media is carried out. Solutions are constructed in the form of multipole series. Theorems on the convergence of series in the topology of compact convergence in R3/0 are proved. The problems on constructing some singular solutions of the theory of vibrations of an anisotropic body are discussed. The fundamental solution of the oscillation equations for an isotropic medium is obtained in a closed form.
138-146
Modeling of Non-Isothermal Viscoelastic-Viscoplastic Deformation of Bending Reinforced Plates
摘要
A model of non-isothermal viscoelastic-viscoplastic deformation of multidirectionally reinforced flexible plates has been developed. The viscoplastic deformation of isotropic materials of the composition is described by the relations of the flow theory with isotropic hardening, which take into account the dependences of the loading functions on temperature and strain rate intensity. The viscoelastic behavior of the composition components is described by the equations of the Maxwell–Boltzmann model. The weakened resistance of reinforced plates to transverse shifts is modeled by the Ambartsumian bending theory relations, and the geometric nonlinearity is modeled in the Karman approximation. The connection between the thermophysical and mechanical components of the problem of inelastic dynamic deformation of reinforced plates is taken into account. The temperature over the thickness of structures is approximated by a 7th order polynomial. The numerical solution of the formulated nonlinear two-dimensional problem is constructed using an explicit scheme of time steps. The viscoelastic-viscoplastic dynamic behavior of a relatively thin glass-plastic plate is studied with and without allowance for the thermal response in it. The structure is loaded transversely with an air blast wave. It is shown that the failure to take into account the thermal response in a fiberglass plate can significantly distort the calculated fields of residual deformations of the components of its composition, despite the fact that the maximum heating of such a structure does not exceed 10°C. Viscoelastic-plastic calculations, when the sensitivity of the composite materials to the rate of their deformation can be neglected, can reasonably be carried out without taking into account the thermal response of the composite plate, if there are no external heat sources of non-mechanical origin.
147-169
Periodic Contact Problems for a Wedge with Friction Forces
摘要
Periodic contact problems for a three-dimensional elastic wedge (a dihedral angle, a half-space and a quarter of space are particular cases), taking into account the Coulomb friction forces in unknown contact areas are considered. One face of the wedge is rigidly fixed, and the other face interacts with an infinite rectilinear chain of identical rigid dies, the axis of the chain is parallel to the edge of the wedge. Friction forces perpendicular or parallel to the edge of the wedge are taken into account. Integral equations are derived in which the series generated by the Cerruti components of the contribution of friction forces are summed exactly. Problems are solved using the method of nonlinear integral equations, which makes it possible to simultaneously determine the contact area and contact pressures. The mechanical characteristics are calculated, the transition from a discrete to a continuous contact area of infinite length is studied.
170-179
К 100-ЛЕТИЮ СО ДНЯ РОЖДЕНИЯ Л.А. ТОЛОКОННИКОВА
180-180