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.

Ограничимся нахождением равнодействующей Т внутренних сил в ннжнем сечении каната, придавая формуле для определения нор­мального растягивающего напряжения о в упомя­нутом сечении значение только приближенной харак­теристики соответствующего напряженного состоя­ния.

Тахограмма подъема предполагается Р. Ф. Ильиным трапецеидаль­ной. Вследствие сложности точного решения им использован прибли­женный метод для малых высот подъема. Задано абсолютное удлине­ние вертикальных частей АХВ и А2В2 каната (Аь А2 — произвольно взя­тые точки каната, В, В2 — точки подвеса грузов) в следующей форме ...

Цель статьи — изучение обтекания плоским потенциальным пото­ком жидкости некоторых алгебраических кривых, а также нахождение профилей аэропланного типа с точкой возврата и указание способа их построения.

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

Применяемый в настоящее время расчет подъемных шахтных кана­тов на прочность носит явно условный характер. В основание этого расчета положен так называемый статический коэффициент безопасно­сти, представляющий отношение разрывающей нагрузки к нагрузке ста­тической (вес каната и концевого груза). Само собой разумеется, что при таком способе расчета динамиче­ская нагрузка совершенно не учитывается и действительное значение коэффициента безопасности (динамического) остается неизвестным. Есте­ственно было ожидать, что указанное положение привело из понятных соображений осторожности к несколько преувеличенным значениям коэффициента безопасности, особенно для глубоких шахт, где вес каната играет существенную роль. В силу этого обстоятельства в американской практике значение упомянутого коэффициента устанавливается диффе­ренцированно, в зависимости от глубины шахты и тем меньше, чем больше упомянутая глубина. По-видимому, условия работы каната в глу­боких шахтах считаются более благоприятными в смысле величины на­пряжений. Следует все-таки указать, что последнее утверждение, хотя и носит на первый взгляд более или менее вероятный характер, до сих пор еще не является вполне обоснованным и, кроме того, имеет исход­ным пунктом оценку напряжений при нормальном режиме подъема.

The problem of determining stresses in hoisting ropes and the related question of longitudinal vibrations of elastic rods 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. Lokshina. 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; it is in matters of mine lifting that it requires further development. Previously, the function e(s), which characterizes the law of change in relative elongation for the part of the rope wound on a drum, was considered arbitrarily specified. Due to the arbitrary assignment of this function, the elements of the above-mentioned part of the rope, passing through position C, undergo an impact at the moment of their inclusion in the vertical part of the BC rope, so the relative elongation and speed of these elements, generally speaking, do not coincide, respectively, with the relative elongation and speed of the part of the BC rope at point C. Note here that the speed values differ from each other by a small amount on the order of relative elongation (see below). The problems of lowering and raising a load lead, generally speaking, to various forms of the second boundary condition. The coincidence of these forms occurs only in one particular case, which requires a special specification 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 rotation of the drum (see 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 mode of the ball mill after some time can be considered steady, and the magnitude and direction of the speed 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 issues of mining and 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 in the theory of screens and conveyors is the study of motion with friction. One of the questions in the theory of screens and conveyors is the study of motion with friction of a particle of material along a plane performing some movement, which in this work is considered predetermined. For this purpose, differential equations of material motion are compiled in the most general case of screen motion and then applied to the case of plane motion, which is of practical interest. Finally, some special cases of integrability of the resulting differential equation are indicated.

The calculation of a cable for strength is in practice somewhat conditional. It is based on the ratio of the breaking load to the static load, which is called the safety coefficient and the smallest value of which is established by the safety rules for lifting. Having set the value of this coefficient and using it to find the number and diameter of the wires , then we perform a verification of the safety factor, partially taking into account the actual stresses in the cable. However, since the latter are not completely taken into account, such a verification cannot give a sufficiently clear idea of ​​the actual value of the safety factor. In the present study we limit ourselves to small lifting heights, bringing the question to the integration of a third-order ordinary differential equation.

The theory of a spiral separator, as well as a similar theory of a screw descent, has been 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 enriched material. The corresponding differential equations of motion were compiled by Prof. M. I. Akimov. Their integration in the first approximation is performed in this article. An essential role in the design of a spiral separator is played by the construction of projections onto the horizontal plane of the trajectories of moving particles (the so-called unloading diagram). The latter allows you to judge how suitable the designed separator is for the enrichment of a certain material. The author set as his task to give another method for constructing an unloading diagram, namely analytical, based on the integration of the differential equations of motion in the first approximation. This latter also makes it possible to highlight the feasibility of new forms of spiral separator proposed by prof. M. I. Akimov. The question of how large the error of the first approximation is 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 row of balls after they were separated from the drum wall to be parabolic. However, some experimenters in their collective work indicate that the balls, during their free movement, are thrown further than is assumed by the theory mentioned above. This circumstance was not taken into account by the theories of White and Davis. Therefore, it seems advisable to reconsider the old theory of the ball mill in order to clarify it and take into account 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 row of balls. Let us note first of all that usually the moment of separation of the ball from the wall of the drum is identified with the moment of separation from the underlying ball. Meanwhile, in reality these moments do not coincide. During the time passing between them, the lower ball continues to exert pressure on the upper one and thereby influences the nature of thetrajectory of the latter.

Let us limit 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, being, as it were, invariably connected with the latter. In the second period, the ball is separated from the wall of the drum, but still continues to rest on the underlying ball, moving along the latter. Finally, the third period begins from the moment the moving ball leaves the lower one. Studying the movement of the ball during the first and third periods does not encounter any difficulties. Therefore, only the second period is the subject of further research. The considered movement of the ball occurs in a vertical plane perpendicular to the axis of rotation of the drum.

Previously, I considered the question of the motion of an ideal incompressible fluid in the presence of a stationary cylindrical solid body, which is relevant to the outline of airplane wings. In this case, the streamlined contour is assumed to be algebraic and the velocity of the fluid at an infinitely large distance from the solid body is constant in magnitude and direction. At infinity, the speed of an ideal incompressible fluid is constant in magnitude and direction. In this paper we show that applying the conformal transformation mentioned above results in an obstacle contour representing an algebraic curve of order 2m (see paper).