Determination of the skip force effect on guides in mine shaft

. An analytical solution is presented to the problem of determining the force effect of lifting vessel (skip) on guides during its movement in the mine shaft. Forces values are obtained using acceleration data from sensors of motion smoothness through monitoring system. The technique developed allows to determine skip force effect on guides along all axes of horizontal coordinate system. A transition from a force to impulse action is provided. The interrelation of force action surges with guides profile deviations is analyzed. The results of this study can be widely used to identify the areas in the mine shaft where emergency could potentially occur.


Introduction
The problem of determining skip force effect on guides is relevant and has not been solved yet. The determination of the skip force effect on guides in real time which can be used in development of special approaches and technologies for preventing emergencies is of particular interest.
Nowadays, a number of researchers are engaged in skip dynamics modeling problem. There are many approaches for solving this problem. So in [1], an analytical solution to the problem of oscillation of a two-mass system was considered. The aim of this study was to reduce the dynamic loads arising in certain modes of skip motion which leads to the occurrence of large oscillations of the lifting vessel. In paper [2], a relationship between the contact interaction of skip with guides and the surfaces of the guides curvature increase is set. The authors of [3] carry out finite element analysis of skip motion taking into account the aerodynamic interaction of the loaded skip and the counterweight skip. As a result, in this paper and previous paper of the same authors [4], the conclusion about the optimal gap between skip has been made. Also, it was noted that the influence of the Coriolis force is insignificant in the model under consideration in comparison with the aerodynamic interaction force of skips. Paper [5] is devoted to the development of the optimal design of buntons that satisfy conditions of strength taking into account force impact from side of skip. The authors of [6] carry out a finite element modelling of the lifting vessel motion taking into account the irregularities of the guides profile. In [7], the influence of fretting on the durability of the hoisting rope taking into account the value of the contact interaction between the rope and the sheave is determined. In works [8,9], an analytical assessment of rope fatigue is made. A number of works [10 -12] are also devoted to the assessment of the processes occurring in the hoisting rope during the movement of the skip. In [13], a mathematical model of skip motion using the Hamilton principle is developed and verified with the data obtained with the use of experimental machine. Hamilton's principle is also applied when modelling the motion of a lifting vessel in [14,15]. Similar mathematical modelling is carried out in [16]. However, the purpose of this work is to assess the effect of the lifting force on the oscillations of the lifting vessel. The case of emergency during the operation of skip is considered in [17]. In particular, the author defines braking parameters that will not lead to critical consequences. The work [18] is devoted to the study of the relationship between the slipping of the hoisting rope during the operation of the hoisting machine and the tension force of the rope. In the article [19], a mathematical model of the skip motion is developed and the analysis of accounting for various parameters on the nature of the oscillations of the studied system is conducted. The article [20] analyzes the skip lift in order to optimize its operation. Also, in this article a new lifting scheme with two lifting friction sheaves is proposed. Modelling of the dynamic behavior of the skip hoist is also described in [21].
After analyzing the literature, one can see that there are many gaps in modeling skip dynamics. Almost all authors of works on this problem agree that the study of skip dynamics is a very difficult task. This is also evidenced by the number of works devoted to this topic and the variety of solution approaches.
This paper provides an analytical solution to the problem of determining the forces arising from the interaction of skip with guides problem using a motion smoothness monitoring system installed on skips of one of the JSC "Belaruskali" shafts. Skip movement during lifting consists of translational motion with a given velocity ( ) v t along OZ axis of the OXYZ fixed coordinate system and five independent motions: two translational motions along OX and OY axes and three rotations around СX ′ , СY ′ , СZ ′ axes of the Koenig coordinate system [22] (Fig. 1). Skip as an absolutely rigid body, with respect to the coordinate system moving translationally vertically with a velocity of ( ) v t , has 5 degrees of freedom and its relative motion under the action of forces ( , ,0)

Equations of skip motion
Equations (1), (2) are vector records of center of mass motion theorem relative to OXYZ coordinate system and change of kinetic moment theorem relative to CX Y Z ′ ′ ′ coordinate system.
We assume that a skip with and without cargo is symmetrical relative to the vertical axis both in geometric and material sense.
We also note that directions of the corresponding axes of CX Y Z ′ ′ ′ coordinate systems and Сxyz differ little from each other due to small curvature of surfaces of the guides.
Hence, skip moments of inertia included in c K ′ and coordinates of points i M in CX Y Z ′ ′ ′ coordinate system are replaced by constant values in Oxyz coordinate system.
We project (1) on the axes OX and OY of a fixed coordinate system: , the equation (2) in projections on the Koenig coordinate system axis has form: It is taken into account that 2 due to the symmetry of the contact points loсation.
Equations (3) -(7) are approximate, since during derivation of this equations, the values of a higher order of smallness were neglected in comparison with Now, having skip dynamics mathematical model, we can proceed to determining the force interaction of skip with guides using the acceleration data coming from the motion smoothness monitoring system.

Determining the force effect on skip from the side of the guides
The velocity V of any point M of the skip relative to the fixed OXYZ coordinate system can be approximately found using formula [24]: Here ( , , ) y z ω ϕ ϕ ϕ =    is the angular velocity of Сxyz coordinate system relative to СX Y Z ′ ′ ′ coordinate system, and the vector CM in СX Y Z ′ ′ ′ coordinate system is replaced as it was done during deriving equation (5) -(7) by its constant value in Сxyz coordinate system.
x y z ε ϕ ϕ ϕ =    Subtracting equality (9) from equality (10), we obtain: We project relation (9) on the fixed , OX OY axes and the equality (11) on axis of СX Y Z ′ ′ ′ coordinate system. Here: ( )  It is necessary to require that det A is not a small quantity to solve the system (13). This can be achieved by choice of a suitable sites for accelerometers installation.
We obtain the main force vector c F and main moment c M of the forces with which the skip acts on guides after solving system (13): The intensity of this force action is determined by the modules of vectors: We obtain formulas for the corresponding impulses instead of formulas (14), (15) The force effect on guides from the side of skip changes when the skip moves and this effect is fixed by functions ( ) The greatest impact on the skip along the entire path of its movement during ascent or descent takes place with a maximum of these functions. In other words, if We exactly mark the positions of the skip where skip force effect on guides is much higher than at other positions according to graphs shown in Figs. 6 -9. Orange lines in Figs. 6 and 7 show the deviation modulus of the guides

τ =
It should be noted that reliability of results may be evidenced by analysis of sections of the greatest surge in forces. The first section of a significant surge of forces (located at the beginning of the ascent) corresponds to the largest deflection of the guides in the frontal direction. The second section of a significant surge of forces (in the middle of the ascent) corresponds to the intersection of the rising skip and counterweight skip which is in good agreement with the results obtained in [3,4].

Conclusions
An analytical solution to the problem of finding the force effect of skip on guides during its movement in mine shaft is proposed. The values of force effect , i d 1,5 i = have been determined using the acceleration values of two skip points obtained with the help of accelerometers of motion smoothness monitoring system. An integral transition from forces to impulse actions has been performed in order to reduce random effects. The results of the solution based on considered model have a certain correlation with profiling data.
It is possible to make conclusions about the section of lifting where an impact on the guides occurs when there is a set of statistics of the force effect surges. This technique can be used in a skip motion monitoring system to prevent emergencies.