Vol 33 Iss. 3

The work aims to find out quantitatively and qualitatively the main circumstances of motion of some vibrating machine intended for transportation and sorting of material, and both pre-resonance and post-resonance modes of operation of this machine are considered.The problem is reduced to integration of a system of linear differential equations with variable coefficients. Integration of the system is carried out by means of decomposition of unknown functions in series by powers of a small parameter.The obtained integrals make it possible to determine the frequencies of free oscillations of the material system under consideration, and hence the conditions of resonance. The totality of the obtained data makes it possible to calculate the strength of the vibrating parts of the structure and to give such ratios of parameters that allow to reduce the angular displacements of its two frames, which are undesirable in the proper functioning of the vibrating machine

The subject of study of the present article is the mechanical side of ball mill operation. It is not our task to consider questions of technological character.For this purpose, we shall analyze the existing theories of the ball mill, beginning with the generally accepted one, developed in the works of Prof. L. B. Levenson and Davis, which has become widespread, giving a number of approximate working formulas justified by practice.Let us proceed to analyze the theory of the ball mill in its essence. In doing so, let us restrict ourselves to the consideration of a cylindrical ball mill consisting of a drum with a horizontal axis of rotation O (Fig. 1) and loaded with a mixture of balls and ore. At a proper angular velocity of rotation of the drum, which is assumed to be constant, and the corresponding value of the mill loading, the following mode is established after some time. The balls, initially rotating together with the drum as one unit, at some point begin their movement relative to the drum and, falling downward, crush the ore by impact. Thus, we are dealing with the established motion of the system consisting of balls and ore. Contact between the latter is accompanied by mutual pressure of the contacting bodies, which disappears during the period of their free parabolic motion.

It is known that if we neglect the vertical acceleration of fluid particles compared to the acceleration of gravity and consider the horizontal components of vertical velocity to be unchanging, then for an approximate solution of the problem of motion of a heavy incompressible ideal fluid, steady uninterrupted pre-critical flow in a channel with a horizontal bottom and vertical walls, we can use the gas-hydrodynamic analogy. This problem is solved, for example, by the method of Academician S. A. Khristianovich for gas flows with subsonic velocities. According to this method, the desired flow is found by means of a plane-parallel flow of incompressible fluid, the boundaries and velocity field of which are close to the boundaries and velocity field of the actual fluid flow in the channel under appropriate conditions. In this paper, we will construct the plane-parallel flow required to study fluid motion in transient smoothly tapering sections of channels with rectilinear vertical walls at the upstream and downstream ends.

_{Let’s consider a homogeneous isotropic prismatic layer of rectangular cross-section under the action of its own weight. Let the layer extend infinitely in the direction perpendicular to the cross-section. Two sides of the cross section are embedded and the other two sides are free (Fig. 1). In this case, the layer will be under plane strain conditions. Let us place the origin at the center of gravity of the layer cross-section and denote: layer height 2c, layer width 2l, weight per unit volume of material g, elastic constants E, n, G.}

The presence of holes in the elements of structures and machines is known to be the cause of redistribution of stresses, which is accompanied by a significant concentration of the latter.The proposed paper investigates the redistribution of stresses on the contour of a hole in an infinite plate bending in its plane. The redistribution is caused by a curvilinear hole with a shape very close to a rectangle. Cutouts with a midline ratio are considered (see article). The problem is treated as a plane problem. The hole is assumed to be relatively small and distant from the outer edges. The boundary conditions on the inner contour are satisfied exactly, on the outer contour, where the influence of the hole is neglected, they are satisfied only approximately.

We consider a plate weakened by a central curvilinear opening, in shape very close to a rectangle with the aspect ratio a:b=2 (a is the vertical, b is the horizontal side of the opening). In the curvilinear contour (Fig. 1), the ratio a:b represents the ratio of the center lines; hereafter we will call this ratio the ratio of the sides for brevity. The plate is subjected to the following loads (see the article). We shall confine ourselves to the study of the contour stresses, which are known to have maximum values, and give general considerations on the zone of influence of the hole. The work is intended to enable designers to use in their practical work a scientifically sound solution.

The mathematical theory of plasticity is based on the law of the existence of a generalized function of the mechanical state of a material at constant temperature and strain rate. The essence of this law is reduced to the recognition for each material of a generalized, independent of the type of stress state, curve of plastic flow in certain coordinates. The conditions of plasticity and strength theory express this law by various equations characterizing the hardening of the material during its plastic deformation.The influence of normal stresses on the resistance of a material to deformation is quite obvious, since normal stresses in shear planes cause frictional forces that facilitate or impede the plastic flow of the material. In order to at least approximate the strength of a material under a complex stress state, it is necessary to perform at least two comparative tests for two types of stress states: tension-compression, tension-torsion or, in extreme cases, tension-cutting. The initial tests should be as simple and reliable as possible.

This paper presents the author's proposed technical method for the calculation of shaft hoisting machine drum shells developed in the thesis. The analytical solution of the questions of calculation and design of drum shells of mine hoisting machines was proposed by B. L. Davydov. However, this general method has not been applied in technical calculations as cumbersome and at the same time approximate, not giving specific design formulas. B. A. Morozov developed the general and on its basis the technical methods of calculation of drum shells of mine hoisting machines (see article). The use by the Novo-Kramatorsky plant named after Stalin of the method of calculating the strength of the submarine hull of Yu. A. Shimansky to calculate the shells of the drums of mine hoisting machines gives approximately the same results as the method of B. A. Morozov. A. Morozov. The above noted disadvantages are largely eliminated in the proposed technical method for calculating drum shells of mine hoisting machines, built on the basis of the general theory of the cylindrical shell using the proposals of B. L. Davydov and V. A. Morozov, verified by practice or experiments, in the part of determining the load on the drum shell.

The Friedrichs' technique of extending a positively defined operator in a Hilbert space to a self-adjoint operator (and hence having everywhere a bounded inverse) is at present apparently the simplest way of proving existence theorems for solutions of boundary value problems for self-adjoint equations of elliptic type. Indeed, according to Friedrichs, the matter is reduced to the proof of an inequality expressing the positive definiteness of the operator in the corresponding Hilbert space, after which the existence of a generalized solution of the problem becomes obvious. At the same time, the very procedure of operator expansion, which has in each case its specific functional-theoretic content, indicates in what sense this generalized solution should be understood. The proposed note aims to show that Friedrichs' result is also valid for positively defined operators acting from one Banach space to another space conjugate to it. As an application, some results on the solvability of elliptic boundary value problems are given. A correlation integral equation is an equation of the following form (see article). The purpose of this paper is to investigate the system of fundamental functions of the equation. We will carry out the reasoning for the case of symmetric correlation (see article). However, this restriction can be easily removed and non-symmetric correlation can be considered if we pass to the system of integral equations using the Hilbert-Schmidt theory. Let us prove a few theorems (see article).

A correlation integral equation is an equation of the following form (see article). The purpose of this paper is to study the system of fundamental functions of the equation. The reasoning will be carried out for the case of symmetric correlation (see the article). However, this restriction can be easily removed and non-symmetric correlation can be considered if we pass to the system of integral equations using the Hilbert-Schmidt theory. Let us prove several theorems (see the article).

Let’s consider a correlation integral equation of the form (see article). All conditions and remarks concerning this equation are fully preserved. To equations of this kind, as it is known, the problem of transforming a curvilinear correlation relation between two continuously distributed random variables into a rectilinear correlation relation between the new variables is reduced. It turns out that the problem of finding the set of all pairs of functions (in the case of symmetric correlation) between which the correlation is straightforward is equivalent to finding the fundamental functions of the equation.

1. Fredholm equation. Let us consider integral equations of the form (1). Finally, we note that the proposed method can be easily generalized to the case of the system of integral equations. 2. Volterra equationThe above idea can be generalized to the Volterra equation of genus II. Let us consider equation (23) (see the article). Thus, equation (23) is equivalent to two functional linear differential equations (33) and (37) with Cauchy conditions (34) and (38), and the solution of equation (23) will be obtained by the formula (see article).

1. Fredholm equation. The solution of the Fredholm equation of genus II with a continuous kernel can be reduced to the solution of two functional differential equations with Cauchy conditions. This fact suggests the idea of constructing solutions of integral equations in the form of degree series. Let us consider the Fredholm equation (see Ref.) 2. Volterra equation. In this paragraph, the above considerations to the Volterra equation. We will consider applying the equation (see article).

Let an integro-differential equation be given (see article). In this paper, we investigate the solution of equation (1) for initial conditions (2) by the method we applied for in [1]. This method, with slight modifications, is easily transferred to the case of m≤n as well.

In [1], [2], [3] we have investigated solutions of some classes of linear integro-differential equations. At the same time, both general solutions and solutions of the Cauchy problem were found for the mentioned types of equations. The method we used in the cited papers can be successfully applied to the solution of the boundary value problem for linear integro-differential equations. Part of the eigennumbers of the integro-differential system (2) at m≤n. are the eigennumbers of the integral equation (32), and another part of them are the roots of the equation D (X) = 0. Together they form the spectrum of eigennumbers of the integro-differential system (2). It follows that the spectrum of eigennumbers of an integro-differential system is a discrete set.

When solving some problems related to the quality of the control process, in a number of cases there is a need to study the distribution of roots of algebraic equations with respect to the rays coming from the origin. The present paper indicates a method of investigating the distribution of roots of equations with respect to regions formed by rays originating from the origin and symmetrically located with respect to the real axis of the plane of roots (Fig. 1 and 2).

The present paper investigates the distribution of large absolute value roots of quasi-polynomials of the form (1) (see article). As a result of the research it is shown that in the presence of the principal term the function (1) has two groups of large modulo roots with negative real parts, and the roots of each group obey the asymptotic dependences established in the work of the author [1]. The questions touched upon in the paper belong to the general problem of investigating the distribution of roots of quasi-polynomials, to which, as is known, many practical problems related to the construction and study of transients in linear systems with distributed parameters and with delay lead.

Barbuti derived some condition for the boundedness of solutions of equation (1) (see article) and showed the relation of this condition to the results of Caccioppoli and Gusarov. Barbuti's results are interesting, but the formulation of the main theorem and the proof are rather cumbersome. Meanwhile, the main proofs can be carried out in a very elementary way, using the usual deductions that are always used in the study of equation (1); they are available, for example, in B. M. Levitan's book “Decomposition by Eigenfunctions".In the present note we give simplified proofs of some results of Barbuti and Gusarov.

Let’s consider the following problem in the theory of heat conduction. An initial temperature distribution is given in a space consisting of two media separated by a flat interface. It is required to find the temperature at any point in the space, at any moment of time. The thermal characteristics of each of the two media are assumed to be constant. The formulated problem has been considered by a number of authors for the one-dimensional case. The possibility of solving the multidimensional case using integral equations was pointed out by Münz [4]. In [5], a two-dimensional problem was solved by the method of successive approximations. In the present paper, a closed-form solution of the problem under consideration is given for the two- and three-dimensional case. The solution method can be applied to a number of similar problems.

In the course of the long historical development of descriptive geometry, as one of the sections of geometry, a number of questions have arisen that currently require scientific research. One of such questions is the question of establishing specific patterns of connection between different in nature methods of design. In graphically solving the various problems of engineering practice, it is necessary to strive for the simplest solution of each problem posed. In a number of cases, the simplest graphical solution of this or that problem can be achieved not only by skillful and reasonable use of one or another design method, but also by the ability to move from one design method to another by the simplest graphical constructions, i.e. to carry out transformation of projections. In the present work it is meant to establish a regularity of connection between the images of elements constructed by their central projection onto a common picture plane, and to show the possibility of using such a regularity in solving some problems of engineering practice.

The moment of inertia of a surface of rotation with respect to the axis of symmetry IX, the abscissa of its center of gravity xc, and the surface area S can be determined, as is known, in this way (see article).In computing the integrals (1), (2) and (3), both in the case of (4) and in many other cases, the technique of uniform approximation of the radical included in the subintegral functions can be used. In order to carry out such an approximation, consider the following problem (see article).

In many practical proposals there is a need to calculate the tangential stresses in hollow shafts in bending. The ratio of inner diameter to outer diameter in hollow shafts is chosen very differently depending on the purpose of the shaft. In cases where the removal of the central part of the cross-section is intended to control the quality of the material, the inner diameter of the shaft is usually small. The formulas available in the literature for determining the greatest tangential stress in hollow shafts in bending usually do not take into account the effect of the diameter ratio on the magnitude of the tangential stress.

The methodology for determining displacements in a semi-rigid coupling with a single corrugation has been described earlier. In practice, two or three corrugations are often used in order to increase elasticity. Increasing the number of corrugations changes the deformation conditions and requires separate consideration. The purpose of this paper is to investigate the effect of corrugations on the displacements of the entire coupling and its individual elements. The main value determining the elasticity of the coupling is the angle of rotation α (Fig. 1). It depends on the value of the bending moment M, the size and material of the corrugation. In the future, we take the angle a, as well as the dimensions of the coupling, as given.The stated methodology for determining displacements in couplings with two corrugations can be extended to couplings with three corrugations.

The successful development of a number of branches of socialist industry, and first of all of metallurgical industry, depends to a large extent on the quality of refractory ceramic products. Modern metallurgy, which uses oxygen blast in blast furnaces and high-speed methods of steelmaking in open-hearth furnaces, needs new, more highly refractory materials than the current ones, whose melting temperature does not exceed 1800- 2000°. The given data (see article) emphasize the great urgency of the problem of new highly refractory materials. The main reason for the difficulties arising in the search for new highly refractory materials is the lack of state diagrams of oxide systems that make up refractory masses. The number of materials from which refractory masses are usually made is rather limited. Therefore, the study of the previously unstudied state diagram of the system MgO - Cr₂O₃ - ZrO₂ (melting point MgO - 2800 °, Cr₂O₃ - 2110 °, ZrO₂ - 2715 °) can have not only theoretical but also practical value in the search for new refractory materials with higher physical and chemical properties. Triple mixtures of magnesium oxide, chromium oxide and zirconium dioxide, whose compositions are located in the region of solid solutions, as not forming eutectics and therefore do not experience softening when heated up to melting temperatures (2200-2600 °), are new, practically important highly refractory materials.