Vol 10 Iss. 1

In this work, which presents a detailed and systematic exposition of the results I obtained earlier,elementary derivations are presented, and a number of new surfaces are indicated that may find application in the theory of the spiral separator and which, with an appropriate choice, offer certain advantages over the ordinary inclined helical surface—both in terms of achieving greater stability of the helical motion occurring on them, and in terms of the possibility of suitably selecting the velocity distribution of the masses moving along them. This work also presents new results concerning the investigation of the stability of the considered motion along a helical line on a rough surface, and clarifies the influence on this motion of: (1) the resistance of the medium, (2) the rotation of the surface about the vertical axis of the helical lines lying on it.

The theory of the spiral separator is based on the study of the motion of a heavy point along a rough helical surface. The article integrates the differential equations of motion of a heavy point along a helical line on a rough surface having a curvilinear generatrix. As Prof. M. I. Akimov points out, this surface can be used for the construction of sorting machines. The motion of a heavy point along this surface occurs with a vertical velocity independent of the helical line of motion.

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.

The refined theory of the motion of the outer layer of balls in a ball mill takes into account the relative motion of an upper ball with respect to a lower ball that is at relative rest with respect to the drum. To determine this relative motion, the refined theory provides two differential equations (see the article). This work is devoted to a quantitative assessment of the new theory. The author of the article studies the motion of the outer layer of balls in a ball mill by observing the motion of the upper ball relative to the motion of the lower ball. Each specific numerical example represents the moment when the upper ball separates from the lower ball.

The subject is fine grinding in ball mills. Two constants are adopted, one of which relates to the structure of the substance and characterizes its disintegration upon impact, the other—the crushing rate under given conditions. We derive a differential equation for the characteristic of the substance, the solution of which is given. We then proceed to consider special cases and ultimately provide a method for determining the constants in accordance with the characteristics of the material.

As is known, Lagrange, in his "Analytical Mechanics", when presenting the theory of small oscillations of a system of points, made the incorrect assertion that when the roots of the characteristic equation are multiple, the solution always contains terms with integer powers of time outside the sine and cosine signs. The fallacy of this assertion was pointed out in 1858 by Weierstrass, who, however, considered only the special case of a system of equations containing terms with second-order derivatives and terms linear in the unknown functions. Routh, in his treatise on the stability of motion, considered the general case of equations also containing terms with first derivatives. This note is devoted to the same question. While containing no essentially new results, it aims to provide a comprehensive study of the conditions under which the so-called secular terms are absent from the solution of a system of linear homogeneous second-order equations of the most general type. The approach is based on Cauchy's method of integrating linear equations—that is, the very method that Weierstrass applied in studying the aforementioned special case. The advantage of this method lies in its naturalness: we seek expressions for the coefficients in the general integral of the system and, by setting them to zero, obtain the desired conditions.

The calculation of the electromagnetic system of a magnetic separator, like that of almost any electromagnetic apparatus or device, consists of two parts: the calculation of the magnetic circuit of the system and the calculation of its winding. This article presents a general method for calculating the windings of magnetic separators, based on the principle of maximum winding efficiency, which allows choosing the optimal values of winding parameters in each individual case. The proposed method can also be applied when calculating other electromagnetic mechanisms (taking into account the specific features of those mechanisms) in cases where the calculation of the windings be carried out independently of the calculation of the magnetic circuit.