Development of a three-parameter model of the shoe brake contact interaction with the drum in mine hoisting machine

For safe operation of mine hoisting machines (MHM) in the mining industry, it is necessary to provide high constructive reliability of brake systems, in particular, brake systems based on the block brake. The contact interaction of the brake with the translational movement of the shoes applied in MHM, remains insufficiently studied. In particular, it is necessary to develop a technique for accurate determination of the braking moment and forces in the elements of the brake linkage, as well as to study the nature of the pressure distribution along the brake beam. The purpose of this article is to simulate the contact interaction of the MHM brake taking into account the coefficient of friction, the ratio of the flexural stiffness to the longitudinal stiffness of a beam, as well as the ratio of the transverse stiffness of a lining to the flexural stiffness of a beam. The application of the developed model will help to reduce the maximum contact pressure in the shoe brakes of the mine hoisting machines and will allow more accurate calculation of the braking moment value compared to existing methods.


Introduction
The hoists are considered to be the most important installation among all the machinery used in the mining industry.An emergency situation in the process of the hoisting machine operation actually stops all the mining works in the mine, which leads to huge financial losses.This determines the high requirements to the technical level of mine hoisting machines and the quality of their manufacturing.
Novokramatorskyi Mashynobudivnyi Zavod (NKMZ), one of the largest production and research complexes in Europe, up to now produces drum-type hoisting machines, which are often equipped with shoe brakes.A lot of well-known scientists have been involved into the designing of the braking device for MHM, but because of the imperfection of the previously used computing facilities, the very important parameters for the interaction between the brake beam, lining and the brake rim, have not been studied.For example, the problem of calculating the brakes of mine hoisting machines has recently been considered by Barecki Z. and Sciezska S.F. [1].However, in their mathematical model of the contact interaction between the brake shoe and the drum, they did not take into account the influence of tangential forces on the distribution of the contact pressure, and the calculation given as an example concerns a particular machine without specifying brake parameters, in addition, the article published on this topic does not contain any generalized dependences.
To calculate the braking parameters for large hoisting machines produced by the NKMZ with the translational movement of the brake shoes, the method described in the works of Davydov B.L., Karpyshev N.S., and Fedorova Z.M. has been used so far.Here, a hypothesis is made about the sinusoidal law of contact pressures distribution, which is consistent with the assumption of the absolute stiffness of the brake beam [3,7,10].
The traditional mathematical model of Davydov B.L. for calculating the distributed tangential and normal forces acting on the brake beam of the MHM shoe brake is based on the assumption that these forces are distributed independently of the lining friction against the drum and the flexural stiffness of the beam.This assumption cannot be considered valid, and therefore the results of determining the specified loads in the construction of a shoe brake can lead to significant errors and the creation of further emergency situations.
The results of the computational experiment carried out by means of the SolidWorks Simulation Program on the study of a shoe brake for the CR-4x3/0.7 type mine hoisting machine with translationally moving shoes, make it possible to construct a U-shaped pressure distribution curve.
Many researchers of contact interaction of shoe brakes with a drum, in particular, Day A.J., Newcomb T.P., Harding P.R., Huang YU.M., Shyr J.S., Tirovic M., Barecki Z. and Sciezska S.F., using in their works the method of finite elements, finite-difference method and the boundary element method, have proved the invalidity of the hypothesis about absolute stiffness of a beam.
The calculations, given in works of Day A.J., with the use of the finite elements method relate to the specific case where the brake beam is modeled as a sequence of beams with different cross sections [4][5][6].
In the calculations of Huang Y.M. and Shir J.S. the boundary element method was used without analysis of the basic parameters influencing the contact interaction [8].
In the works of Zabolotnyi K.S., Kolosov D.L., Moskalova T.V. the techniques were described for the construction optimization of drums and brakes in the mine hoisting machines by determining the stress-strain state [11, 15, 17 -20].
In further numerous works on improving the calculation technique for brakes, the dynamic and thermal processes were taken into account which proceed in the braking system elements.For example, in the works of Kyrychenko Y.O., Samusia V. I., Ilyn S.R. and Bondarenko V.I., the problems of hoisting plant dynamics during working and safety braking was considered.Most importantly, that the hypothesis of stiffness of the brake rim and beam, was not substantiated [2, 9, 12 -14, 16].

Development of a three-parameter model of contact interaction
Fig. 1 shows the scheme of a part of the braking system for large-size MHM produced by NKMZ.By means of the brake linkage, the braking force is transferred to the upper 4 and lower 5 rods.The center of the brake beam rotates on a vertical pillar 3. The brake shoe consists of a brake beam 1 and a set of linings 2. These linings interact with the drum 6 during braking.Thus, it can be assumed that the brake shoe is affected by two couplings: a vertical pillar and a drum.
The exact calculation of the forces arising in the brake shoe is possible only with the use of numerical methods (the methods of finite or boundary elements), but in order to reveal the basic laws of contact interaction, it is required to create a simplified analytical model of a structure.It has been decided to represent it in the form of a circular bar with the constant cross section, fixed from vertical displacements in the central point.According to the accepted assumption, the same external forces act on the bar in the horizontal rods.
The physical model of the brake lining has been defined as an array of elastic bodies, like Winkler foundation, which do not interact with each other, work in compression and transfer the distributed frictional forces arising between the brake drum and the brake shoe, calculated for the limited equilibrium state in accordance with Coulomb's law (Fig. 2).The condition has also been accepted that the drum rotates clockwise, and the angle of wrap of the brake shoe is equal to 2γ.Having replaced the couplings on reactions, we determine the concentrated vertical force y F in the central point of the bar, as well as the distributed forces q and p of the drum action on the lining.In view of small thickness of the beam, we consider that these forces are applied to the neutral axis of the bar.The action of the horizontal movable bearing in the brake system causes the occurrence of a force jump (vertical reaction of the bearing), which ensures the discontinuity of the longitudinal force function, excluding the possibility to consider the bar as an element of structural mechanics.Therefore, in the model for calculation, it is provided that the bar consists of two parts: the upper and lower ones.
The mathematical model for calculation of forces in the brake shoe includes the number of the equations presented below.
Equilibrium equations for the lower ( i = 0) and upper ( i = 1) parts of the bar have the following form: where ,1 , T i (φ) is the longitudinal force; Q i (φ) is the shear force; M i (φ) is the bending moment; φ is the current angular coordinate; R is the radius of the neutral line of the brake beam.
The equation for calculating the distributed normal forces that correspond to the model of a structure with an elastic base, can be represented as follows: where w i (φ is the depth of beam deflection; E l is the elasticity modulus of the lining material; H l is the thickness of the lining; B b is the width of the brake field; κ is the transverse stiffness of the lining, defined by the following expression: The equation describing the distributed friction force according to Coulomb's law has the following form: We formulate Hooke's law for determining the bending moment, that is where E is the elasticity modulus of the beam material; I is the inertia moment of the beam section; θ i is the angle of the beam rotation.
Here we take into account the following kinematic dependence: and Hooke's law for determining the longitudinal tension, that is where F is the cross-sectional area of the beam.Both bars are connected in a place designated in the scheme by the point K, in which the value ϕ = 0, they are fixed there from vertical displacements.Thus, the state of the construction corresponds to the following dependences: ( ) At the upper and lower ends of each beam, zero values of the bending moments are set, as well as the tangential and radial components of the horizontal forces in the rods, that is Now we turn to the determination of dimensionless quantities, namely: is a multiplier factor proportional to the applied force.
To reduce the volume of the record, we transform the listed expressions into a vectormatrix form.
The vector of undetermined values is a transposed line, that is: Then the equation corresponding to the mathematical model will be as follows: where

EI ζ = EFR
To solve the formulated system of linear differential equations with constant values of the coefficients, Euler's method is used.When used for the MHM braking system, the following dependences are valid: where n is the eigenvalues of the matrix А; Ф is the eigenvectors of the matrix А.
If the coefficient of friction is zero, then the zero root becomes reiterated, namely: This makes it necessary to find out the eigenfunction as a linear one, that is ( ) The solution of the differential equations system corresponds to a linear combination of fundamental functions with undetermined values of the coefficients, that is To determine these coefficients, the boundary conditions given below (Тable 1) are used.These conditions are assigned in the form of the following system of algebraic equations: .

GS = B
To calculate the coefficient values, the function S = lsolve (G, B) built into the MathCAD program is used.
We designate the ratio of the contact pressure to the pressure caused by the same force N acting on an absolutely rigid section of the same area (F = BR2γ) by the adjusted pressure value Ξ.We denote by χ its maximum value.
Table 1.Boundary conditions of the mathematical model for determining the forces acting on the brake beam.

Boundary conditions Physical meaning
( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 shows the distribution curves of the adjusted values of the contact pressure according to different values of the parameters λ and ζ.Here we see that the parameter ζ influences the pressure distribution only in case when λ > 100.The nature of the distribution depends essentially on the relative transverse stiffness of the structure.Thus, if λ > 10, then the distribution curve has a U-shaped form with a clear-cut edge effect, but in case of λ value increasing it becomes asymmetric.When λ ≤ 10, the character of the curve is sinusoidal, according to which the maximum values of the contact pressure are observed in the center of the brake shoe.Close distribution to the uniform distribution of contact pressures, taking into account the considered values of the parameter ζ occurs if λ ≈ 10.Neglect of the frictional forces when calculating the distribution of the contact pressure leads in the sequel to an error of the obtained maximum pressure values of 1.3%, if ζ = 0.002, and 11% when ζ = 0.07.
In order to consider both types of distribution, we use the above described adjusted maximum value of the contact pressure Ξ.Since in Fig. 3 the adjusted values of the contact pressure in specific points are indicated, and the value of the parameter λ exceeds the value of the parameter ζ, we plot the graph of the maximum pressure dependence as a continuous function λ (Fig. 4).To carry out the numerical experiment, the characteristics of the brake beam of the CR-4x3/0.7 machine with the following parameters have been chosen: -radius of the neutral line of the brake beam R = 2.26 mm; -width of the brake field l B = 400 mm; -half of the grip hold arc of the brake drum γ = 50º; -thickness of the brake beam H = 400 mm; -thickness of the lining l H = 80 mm; -elasticity modulus of the beam material E = 2.1•10 11 Pа; -elasticity modulus of the lining material l E = 3•10 8 Pа; -coefficient of friction f = 0, 0.3; -relative stiffness of the brake beam λ = 109.4;-relative flexural stiffness of the brake beam ξ = 2.61•10 -3 .The graph of the contact pressure calculation results is shown in Fig. 6, where the red color indicates the curve reflecting the calculation by the model without consideration of the effect of friction on the contact pressure distribution (hereinafter 'without friction'), the blue color -the curve with account of frictional forces and the green color indicates the curve constructed by calculations using the model of Davydov B.L. Having transformed the equations describing the distributed normal forces in the beam, we obtain the following dependences: -braking moment: ( ) As in the previous case of the mentioned models use, the force in the vertical pillar has the following values: 2.75•10 5 N; 3.48•10 5 N; 2.75•10 5 N respectively.The error in determining this indicator with the use of both models, simplified and Davydov B.L., is 26.5%.cut edge effect.In this case, the value of the maximum contact pressure, calculated by the model of Davydov B.L., has turned out to be by 2.5 times lower.
The error in the braking moment determination with the application of the model of Davydov B.L. is 11.3%, and the efforts in the vertical pillar is 26.5%.
Authors express gratitude to Yurii Ovchynnikov, the Lead Designer of the Novokramatorskyi Mashynobudivnyi Zavod for providing help and consultations during the work performance.

Fig. 1 .
Fig. 1.Scheme of a part of the MHM brake system.

Fig. 2 .
Fig. 2. Scheme of the forces arising in the brake beam.

Fig. 5 .
Fig. 5.The level curves of the adjusted contact pressure maxima depending on different λ and ζ parameters values.

Fig. 6 .
Fig. 6.Graph of the contact pressure values distribution along the arc of the brake beam.
The force value in the horizontal rods applied to this machine is 699 kN, used in the calculations based on the model of Davydov B.L.[3].Then the braking moment calculated with the use of 'no friction' model is equal to 7.72•10 5 N•m, with account of friction -7.87•10 5 N•m, but the braking moment calculated by the model of Davydov B.L. -6.98•10 5 N•m.The deviation of this parameter value towards the decrease when calculating with the use of simplified model is 1.94%, but in the calculations by the model of Davydov B.L. -11.3%.
where λ is the relative transverse stiffness, defined as follows: 2.