Forces at the point of the rope running over the drum. To consider the scheme of lifting the load, we introduce the notations: ...

The condition of continuity (continuity) was first given for the general case in analytical form by Euler, and then later in a different form by Lagrange.In this paper, the motion of the medium is described by means of Euler variables, which are related by Taylor series with Lagrange variables. The first part of the paper relates to a one-component fluid, the second to a two-component fluid.

Consider the plane vortex-free motion of a boundless fluid in the presence of a stationary cylindrical solid. The fluid is assumed to be ideal and incompressible, and its velocity at an infinitely large distance from the solid body is assumed to be constant in magnitude and direction. Let us restrict ourselves to the case of continuous steady motion.

The theory of the ball mill was basically developed by Davis in the first quarter of the present century after an experimental study of the ball mill. Much later corrections and additions were made to it. At that time the question arose of taking into account the factor of pushing by the underlying balls of those balls which had already separated from the wall of the drum and were making their relative motion in it, forming a chain of balls disintegrating at the top. Due to the error made, the researchers did not give a correct solution to this question. The error was eliminated and the true shape of the chain of balls was established somewhat later. However, this error is also found in the new foreign literature, so it is reasonable to return to this issue once again and to outline the kinematics and dynamics of the ball mill from the point of view of the theory of chains of balls (refined theory of the ball mill) and give a critical assessment of the latter.

Among the works of Prof. M. I. Akimov, his thesis “On Bessel functions of many variables and their applications in mechanics” occupies the first place in importance.

Let us limit ourselves to finding the equidistance T of the internal forces in the bottom section of the rope, giving the formula for determining the normal tensile stress o in this section the value of only an approximate characteristic of the corresponding stress state.

The tachogram of the ascent is assumed by R. F. Ilyin to be trapezoidal. Due to the complexity of the exact solution, he used an approximate method for small lifting heights. The absolute elongation of the vertical parts A₁B₁ and A2B2 of the rope (A₁, A₂ - arbitrarily taken points of the rope, B₁, B₂ - points of suspension of weights) in the following form is set ...

The aim of the paper is to study the flowing of some algebraic curves by a plane potential fluid flow, as well as to find aeroplane-type profiles with a return point and to indicate the method of their construction.

Application of Poisson's solution. The problem under consideration has been the subject of study by many authors, but, despite the results achieved, is still quite far from its complete resolution, especially from the point of view of the efficiency of the applied methods. Meanwhile, for practical applications this side of the matter is very significant. Therefore, the work of V. A. Staroverova, devoted to this question, should be recognized as very relevant.

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.

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

Currently used calculation of hoisting mine ropes for strength is clearly conditional in nature. This calculation is based on the so-called static safety factor, which represents the ratio of the breaking load to the static load (weight of the rope and end load). It goes without saying that in this method of calculation the dynamic load is not taken into account at all and the actual value of the safety factor (dynamic) remains unknown. It was natural to expect that the above provision led, for obvious reasons of caution, to somewhat exaggerated values of the safety factor, especially for deep mines, where the weight of the rope plays a significant role. Because of this circumstance, in American practice, the value of the said coefficient is set differentially, depending on the depth of the mine and the smaller the greater the said depth. Apparently, the working conditions of the rope in deep mines are considered more favorable in terms of the magnitude of stresses. It should still be pointed out that the latter statement, although more or less probable at first glance, is still not fully substantiated and, moreover, has as its starting point an assessment of stresses in the normal lifting mode.

The problem of determining stresses in hoisting ropes and the related question of longitudinal vibrations of elastic bars in terms of formulation and solution methods have a long history. The present work aims to continue the analytical development of the problem of determining stresses in hoisting ropes of variable length in order to supplement the theoretical material necessary for rational strength calculations of hoisting ropes. The calculation itself is carried out by combining the most unfavorable circumstances in terms of strength, taking into account both the normal lifting mode and its special cases. The methodology of such calculation requires additional research and is not included in our task.

At higher lifting heights, both the mass and the weight of the rope must be taken into account. We will consider the latter as an elastic thread, neglecting its transverse dimensions, and limit ourselves to the case of a rope of constant cross-section. Thus, we come to some particular problem of mathematical physics.

The problem of determining stresses in lifting ropes and the question of longitudinal vibrations of elastic rods, related to it in formulation and methods of solution, have a long history, set out in a recently published monograph by Prof. A. S. Lokshin. As early as 1815, Poisson studied the free vibrations of a prismatic rod with a weight at the end. Without setting out to list all the works that were closest in time, we note Boussinesq’s research on the impact of a body of a certain mass on a conical rod of infinite length, as well as the memoirs of Saint‑Venant, who studied the impact of two prismatic and conical rods. The same authors also considered, among other related issues, the question of longitudinal vibrations of a rod with a load at the end. The main results of these authors are presented in the famous book by A. Love.

The problem of mathematical physics considered in this work is of great importance in mining, specifically in matters of shaft hoisting, and requires further development. Previously, the function e(s), which characterizes the law of variation of relative elongation for the part of the rope wound onto the drum, was considered arbitrarily assigned. Due to this arbitrary assignment, the elements of the aforementioned part of the rope, when passing through position C, undergo an impact at the moment they are incorporated into the vertical part BC of the rope, so that the relative elongation and velocity of these elements do not, generally speaking, coincide with the relative elongation and velocity of the part BC of the rope at point C, respectively. Note here that the values of the velocities differ from each other by a small amount of the order of relative elongation (see below). The problems of lowering and raising a load lead, generally speaking, to different forms of the second boundary condition. The coincidence of these forms occurs only in one special case, which requires a specific assignment of the function e(s). Finally, in conclusion, we examine as an example the nature of elastic deformations of the rope for the initial period of drum rotation (see the article).

In a previous work devoted to the theory of a cylindrical ball mill, I studied the initial period of unsteady motion of the balls. If the angular velocity of rotation of the drum remains constant, then the regime of the ball mill after some time can be considered steady, and the magnitude and direction of the velocity of different balls passing through one and the same point in space do not change over time. We will talk about the outer row of balls. The same reasoning applies to other rows, if we make the assumption that the relative motion of the different rows is independent. This assumption, of course, requires additional research. The ball mill theory, which is usually used in practice, is approximate. The subject of this work is to refine the usual approximate theory of a ball mill.

The problem is to study the motion of a particle moving under the influence of gravity and in the presence of friction along the outer surface of a circular cylinder rotating around a horizontal axis. The problem under consideration finds application in some aspects of mineral processing, namely in the theory of a drum separator, used for ore enrichment, as well as in the theory of a drum dumper, which is intended to unload material moved by a belt conveyor at a certain point. The results obtained make it possible to clarify the above theories, which are usually presented in technical manuals.

One of the questions of the theory of screens and conveyors consists in the study of the motion, with friction, of the particles of some material on a surface which also performs a certain movement, the latter being indicated beforehand. For this purpose, one has to form differential equations concerning the motions of materials in the most general case of the movement of the screen and then to apply them to the case of plane motion, which is of practical importance. Lastly, there are indicated some particular cases of integrability of the differential equation obtained.

The strength calculation of a rope is in practice somewhat conditional. Specifically, one proceeds from the ratio of the breaking load to the static load, which is called the safety factor and whose minimum value is established by safety regulations for hoisting. Having set this factor and determined the number and diameter of wires with its help, a verification of the safety factor is then carried out, partly taking into account the actual stresses in the rope. However, since the latter are not fully accounted for, such verification cannot provide a sufficiently clear picture of the actual value of the safety factor. In the present study, we confine ourselves to small lifting heights, reducing the problem to the integration of an ordinary third-order differential equation.

The theory of the spiral separator, as well as the analogous theory of the helical chute, is very little developed. Therefore, it should be noted that Prof. L. B. Levenson attempted to carry out a constructive calculation of the separator based on an analysis of the movement of the material being processed. The corresponding differential equations of motion were formulated by Prof. M. I. Akimov. Their integration in the first approximation is carried out in this article. A significant role in the design of the spiral separator is played by the construction of the projections onto the horizontal plane of the trajectories of moving particles (the so-called discharge diagram). The latter allows one to judge how suitable the designed separator is for the concentration of a particular material. The author set as his goal to provide another method for constructing the discharge diagram, specifically an analytical one, based on the integration of the differential equations of motion in the first approximation. This latter method also makes it possible to clarify the practicality of the new forms of the spiral separator proposed by Prof. M. I. Akimov. The question of how large the error of the first approximation is constitutes our further task.

The motion of balls in a ball mill was the subject of research by White (1905) and Davis (1920), who considered the trajectories of the outer layer of balls after their separation from the drum wall as parabolic. However, some experimenters, in their collective work, point out that during their free motion, the balls are thrown farther than is assumed by the aforementioned theory. This circumstance was not taken into account by the theories of White and Davis. Therefore, it seems advisable to re-examine the old theory of the ball mill in order to refine it and to account for the mutual pressure of the balls during their relative motion with respect to the drum. The subject of our next study will be the trajectory of the outer layer of balls. Let us first note that the moment of separation of a ball from the drum wall is usually identified with the moment of its separation from the ball lying below it. In reality, however, these moments do not coincide. During the time interval between them, the lower ball continues to exert pressure on the upper ball and thus influences the nature of the trajectory of the latter.

We shall confine ourselves to considering a ball mill consisting of a cylindrical drum with a horizontal axis of rotation and loaded with a mixture of balls and ore. The motion of a ball in a ball mill can be divided into three periods. During the first period, the ball has no relative motion with respect to the rotating drum and moves as if rigidly connected to the latter. In the second period, the ball separates from the drum wall but still rests on the ball below it, moving along the latter. Finally, the third period begins at the moment when the moving ball leaves the lower ball. Studying the motion of the ball during the first and third periods presents no difficulties. Therefore, the subject of further investigation is only the second period. The motion of the ball under consideration takes place in a vertical plane perpendicular to the axis of rotation of the drum.

Previously, I considered a problem related to the shape of airplane wings, concerning the motion of an ideal incompressible fluid in the presence of a stationary cylindrical solid body. In that case, the contour being flowed around is assumed to be algebraic, and the fluid velocity at an infinitely large distance from the solid body is constant in magnitude and direction. At infinity, the velocity of the ideal incompressible fluid is constant in magnitude and direction. In this article, we show that the application of the aforementioned conformal transformation leads to a contour of the obstacle representing an algebraic curve of order 2m (see the article).