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Mine hoist is an important piece of equipment in mine hoist systems, and we achieve deep mine hoist through the multilayer winding, but the cable always undergoes severe shock and vibration during the winding process, and the dynamic load and wear would greatly reduce the lifetime of the cable and cause potential safety hazard. In this paper, we start from the course of crossing over of winding cable, use the methods of differential geometry, mechanics, and mathematical analysis, study the movements of the crossover, and derive the important formula that can reduce the vibration of cable during the course of crossover: the formula about central angle of the crossover arc. The results display that four factors contribute to central angle of the crossover arc, that is, the gap of the rope grooves, friction coefficient of the cable, and diameter of the drum and the cable. The result can provide valuable information for designing multilayer winding mine hoist.

Along with the exploitation of the deep earth resources, the deep mine hoisting is more and more important. Thus, the most common winding system is the multilayer winding drum installations in mine hoisting applications. Then the external wear and plastic deformation are serious at the flange of the drum and the crossover zone [

The winder drum surface is covered by parallel circular grooves with two diametrically opposed crossover zones per drum circumference, and the mechanism applied on the winder drum surface in order to achieve a uniform coiling pattern, as shown in Figure

Expanded view of the drum with parallel groove.

At present, the problem of the crossover transition in multilayer winding is mainly on the way of differential geometry and mathematical analysis. Then the study is mainly about the fiber coiling onto the solid of revolution. In the technology of fiber’s coiling, the geodesic is adopted in coiling in the early stage because of steady and simple calculating. He [

The friction exists between the cable and the drum or among the cables of different layers, and this kind of friction could resist a certain degree of slip. So the cable could remain stable even though it diverges from the geodesic within limits, and we can derive the equation of the crossover curve with the theory of stable coiling along the nongeodesic. Gong et al. [

In this paper, we will utilize the theory of the fiber stable coiling onto the solid of revolution along the nongeodesic and use the thinking and methods of Gong et al. [

When one selects the diameter of the drum, the bending stress of the cable should not be large, and then the cable could keep a certain carrying capacity and lifetime. When the ratio of

In order to derive the reasonable length of crossover zone, the differential geometry relationship between the upper layer and the under layer in the crossover zone must be clear. The following fundamental assumptions are made:

Plan figure of the crossover.

Sectional view of the crossover.

The graphic model of the cable of crossover.

The two formulas can be linearized (ignore the infinitesimal higher than the second order) and get the differential geometry relation of crossover as shown in

As shown in Figures

Equation (

Equation (

When the cable of the second layer moves from one groove to another through the top of the first-layer rope, the cable of the second layer is in balance under the action of tension, friction, and extruding force, as shown in Figure

The relational graph on the geodesic curvature

In the formula

The curved surface equation determines the geodesic curvature

The cable of the first layer winds on the drum which installed parallel groove with two crossover zones, and the cable only shifts a half diameter of one cable along axis, and the helical angle is very small and can be looked at as zero, so the cable of the first layer can be looked at as torus approximate. The parameter equation of the torus can be derived through Figure

The sketch map of each parameter of the torus.

The hook face

The Euler equation is given as follows:

The results which had been simplified are substituted into differential equation (

In order to solve differential equation (

Then differential equation (

In order to get the particular solution of the differential equation of the first order, the initial value must be given. The numerical value of relevant parameters can be given based on The South African Bureau of Standards 0294 (2000) [

The sketch map of the crossover.

So the particular solution of differential equation can be gotten as follows:

The particular solution is substituted into (

Equation (

Equation (

The graphics of the functional relationship of

Figure

For explaining the question well, consider an ascending cycle of a deep mine system with the fundamental parameters; that is,

In the second example, Borje of ABB Mining [

As another example, Kaczmarczyk and Ostachowicz [

It is not hard to find that the coil crossover arc this paper derived is always bigger than the project data in deep mine hoist. As for why these happen, the parameters that the literature [

Contrasting this with earlier Lebus liner, Wieschel [

From another point of view, when the cable achieves multilayer winding, the cable would shift the rope loop in the crossover zone; the abrasion and friction in the crossover zone are much severer than in the parallel circular groove and seriously affected the cable’s lifetime. Thus, the crossover zone is much longer, the more the cables that would participate the shift in the crossover zone, the more the cables that would wind without groove, and this leads to the cable’s instability and irregular arrangement, and this aggravates the wear of the cable and decreases the lifetime of the cable. On the contrary, if the length of the crossover zone is too small, the time that the cable shifts through the crossover zone is too short; it will produce instantaneously shock and greatly shorten the life of the cable. This paper only considers the cable to not slip in the crossover zone. But this paper still gives a better theory reference on the study of multilayer winding in mine hoist.

In this paper we have established the cable’s differential geometry relation in the crossover zones and derived the mathematical expression of the crossover arc. By way of analyzing the differential geometry relation and the equilibrium relationship of the separate forces, the relational expression between geodesic curvature

The formula this paper derived shows that the central angle of the crossover arc of the deep mine hoist is about the four factors, that is, the diameter of the drum, the diameter of the cable, the friction coefficient of the cable, and the gap of the groove. The relationships among them are shown in (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The study is supported by the National Basic Research Program of China (973 Program 2014CB049403).