Study of a Reamer for Raise Workings

I. V. Rodionov, Yu. A. Rozenberg, V. N. Karminskii

Introduction

As studies of various methods of driving raise workings in the Kuzbass have shown [1], modern blasthole-drilling machines of the SBM-3u and LBS-2 types, as a rule, do not ensure production of a completed working of the required cross section—1.5 m. One of the main reasons for this is the absence of a tool by means of which it would be possible, with the present power of blasthole-drilling machines, to ream boreholes from 300–390 to 1200–1500 mm. At present numerous attempts are being made to create the necessary tool. Considering the difficulty of experimental work on testing new tools in a mine, work on creating new working members for blasthole-drilling machines must be accompanied by preliminary theoretical investigations. The purpose of such investigations should be to determine the trajectories of the cutters, the geometry of the tool, the cutting forces, the drive power, and the reactive forces.

Below is presented a study of one variant of the design of reamers of the RUP-1 type, proposed and developed by the Department of Mining Machines and Mine Transport of the Tomsk Polytechnic Institute¹.

Brief Description of the Design

The reamer for coal chutes RUP-1 (Fig. 1) is intended for reaming boreholes 390–500 mm in diameter, drilled in coal by the SBM-3u machine [2], to a diameter of 1500 mm. Destruction of the face is carried out by two crowns I and II, each having four cutting teeth (cutters). The drive for the reamer is the SBM-3u machine, connected to it by means of a drill string. The last rod 14 of the string is screwed into

threaded end of the shaft 15 of the reamer. On the reamer shaft, a straight-tooth bevel gear 1 and a straight-tooth cylindrical gear 5, interlocked with one another, are mounted on a key. Gear 1 transmits rotation to both cutterheads through two identical transmissions consisting of bevel gears 1, 2, 3, and 4. Gear 5 is the central driving gear of a two-stage planetary mechanism and meshes with two planet gears 6. On the axes of the planet gears 6, the planet gears 7 are mounted on keys; the latter, in turn, mesh with the central fixed gear 8. The stationary position of ring 8 is ensured by the brake lantern 10 with shoes 11, which, when the reamer is moved along the axis of the borehole, cut into its wall by means of cutters 12. All the gear transmissions of the reamer are enclosed in a common housing 9. The latter, as the planet gears 7 run around the fixed ring gear 8, turns together with the cutterheads about the axis of the borehole and thus performs the role of a guide.

Fig. 1. Kinematic diagram of the mechanism of the RUP-1 reamer:
\(D_1 = 390\) mm—the diameter of the initial borehole; \(D_2 = 1500\) mm—the diameter of the enlarged borehole.

A cutterhead with teeth (cutters) has a complex motion, consisting of:

1) translational carrying feed motion along the borehole axis together with the drill string, at the speed of the working stroke-

which is provided by the SBM-3u machine during reaming,
\(V_{\text{feed}} = 6.96\ \text{cm/min}\);

2) a translational rotary motion about the axis of the borehole together with the housing of the reamer reducer;

3) a relative rotary motion about its own axis in a plane parallel to the diametral plane of the borehole and spaced from the latter by a distance \(b = 120\ \text{mm}\).

The bits have such a direction of rotation that the coal rolls off the face surface in the direction from the periphery toward the center of the borehole, which facilitates the transport of the coal and reduces the reactive torque about the axis of the working. The direction of reaming is ensured by the previously drilled borehole, along which the drill string with guide lanterns and shoes moves. The latter have surfaces that ensure a definite position of the reamer relative to the walls of the coal chute.

The enlargement is carried out in the following order. After a borehole of diameter 390–500 mm has been drilled by the SBM-3u machine, the drill bit and the straight-feed reamer are removed from the string, and the RUP-1 reamer is installed above it in the expanded state.

To prevent rotation of the brake lantern \(10\) and the crown planetary wheel \(8\), a rod is inserted into the eye of cover \(16\), which is fixed between the uprights in such a way as to ensure movement in the axial direction. After the shoes \(11\) with cutters \(12\) have penetrated into wall \(13\) of the chute, the rod is removed, a rope is fastened to the eye, and the process of reaming the chute continues. Upon completion of the reaming, the reamer is disconnected from the drill string, the bits are removed, and the shoes are folded into the transport position. In this form the reamer, with the aid of the rope and winch, is delivered to the ventilation drift and transported to the next borehole. If necessary, the reamer can be disassembled into three parts: the reducer, the crown, and the bits.

Kinematic study of the RUP-1 mechanism

Determination of the angular velocities of the mechanism links

The number of revolutions of the driving bevel gear \(1\) of the bit drive and of the driving central wheel \(5\), blocked with it, of the main planetary mechanism is equal to the number of revolutions of the spindle of the SBM-3u machine during reaming and is equal to \(n_1 = n_5 = 51\ \text{rpm}\).

We determine the absolute angular velocities of the links of the main planetary mechanism by the tabular method according to the Swamp rule [3, 4]. As a result we have: the number of revolutions of the cor-

of the gearbox housing, which acts as the carrier when planetary gear 8 is stationary, is equal to

\[ n_{\text{red}} = n_{\text{c}} = \frac{n_6}{1+\dfrac{z_8}{z_7}\cdot\dfrac{z_6}{z_5}} = 3.86\ \text{rev/min}. \]

The angular velocity of satellites 6 and 7 is

\[ n_6 = n_7 = n_{\text{c}}\left(1-\frac{z_8}{z_7}\right) = -15\ \text{rev/min}, \]

where

\[ z_5 = 20;\quad z_6 = 50;\quad z_7 = 18;\quad z_8 = 88. \]

The minus sign indicates that the direction of rotation of satellites 6 and 7 is opposite to the direction of rotation of central gear 5 and of the gearbox housing about the borehole axis.

Fig. 2. Determination of the angular velocity of the bit relative to the reamer housing.

To determine the angular velocity of the bit relative to the housing of the guide gearbox, we decompose the complex motion of the bit into two simple motions (Fig. 2): a) rotation of the bit with driving central gear 1 and a stationary carrier; b) rotation of the bit with gear 1 stationary and with the gearbox housing rotating about the borehole axis at \(n_{\text{c}} = 3.86\) rev/min. The resultant angular velocity of the bit is

\[ n_{\text{k}} = n_1 = (n_1 - n_{\text{c}})\frac{z_1}{z_2} = 94.28\ \text{rev/min}, \]

where

\[ z_1 = 36;\quad z_2 = 18. \]

Analytical study of the kinetics of the cutting edge of a crown tooth and the equation of the bottom-hole surface

Let us choose a fixed rectangular coordinate system with its center at the point \(O\) of intersection of the axes of the borehole and of the gear wheels 2 and 3 of the reamer reducer. We shall direct the \(z\)-axis upward along the borehole axis, and place the \(x\) and \(y\) axes in the perpendicular plane. Moreover, we shall make the \(Ox\) axis coincide with the axis of the gear wheels 2—3 of the drive of crown I (Figs. 1 and 3). Then, at \(t=0\), the vertical

Fig. 3. Operating diagram of the reamer:
\(a\)—horizontal projection; \(b\)—vertical projection.

the plane passing through the extreme points of the cutting edges of the teeth of the right-hand bit I will be parallel to the plane \(Oxz\), and \(C_0D_0\), equal to \(2r\) (\(r = 295\) mm). In Fig. 3 the initial position of the reamer reduction gear \(OA_0B_0\) and the right-hand bit \(B_0C_0D_0B_0\) is shown schematically (\(OA = a = 446\) mm; \(AB = b = 120\) mm). At the initial moment, the position of the cutting edge of the first tooth \(3^1_0\) coincides with point \(D_0\).

Let us consider the motion of the most distant point \(3^1\) of the cutting edge of the first tooth of bit I. Point \(3^1\), together with the bit, is in a compound motion which, as stated above, may be decomposed into three simplest motions.

Denote:

\[ \varphi = \omega_p t \]

—the angle of rotation, during time \(t\), of the entire system in the transport rotary motion about the \(z\)-axis;

\[ \omega_p = \frac{\pi n_p}{30} = 0.404\ \text{s}^{-1} \]

—the angular velocity of the transport rotation;

\[ \Theta = \omega_k t \]

—the angle of rotation of the bit about its own axis during the same time \(t\);

\[ \omega_k = \frac{\pi n_k}{30} = 9.868\ \text{s}^{-1} \]

—the angular velocity of the relative rotation.

Then

\[ \Theta = 24.42\varphi = i\varphi . \]

Suppose that during time \(t\), in the transport rotation, the reamer reduction gear together with the bit has turned about the \(z\)-axis through the angle \(\varphi\) and has taken the position \(OABCD\). In this case the edge of tooth \(3^1\), in relative motion, rotating about the bit axis, will traverse the angular path \(\Theta = \Theta_{\mathrm{II}} = 24.42\varphi\) and will occupy the position \(3^1_{\mathrm{II}}\) (Fig. 3, b). The position of the cutter edge \(3^1_{\mathrm{II}}\) in the plane \(Oxy\) is determined by the segment \(D3^1_{\mathrm{II}}\) (Fig. 3, a), equal to the segment \(3^1_0m_{\mathrm{II}}\) (Fig. 3, b).

In general form one may write

\[ D3^1_i = 3^1_0 m_i = r - r\cos\Theta . \]

The formula is valid for any value of the angle \(\Theta\).

From the drawing (Fig. 3) we determine the coordinates \(x\) and \(y\) of point \(3^1_{\mathrm{II}}\). After transformation we shall have:

\[ x = (a + r\cos\Theta)\cos\varphi + b\sin\varphi; \]

\[ y = (a + r\cos\Theta)\sin\varphi - b\cos\varphi. \]

The equations obtained represent the laws of motion of the edge of tooth \(3^1\) in the plane \(Oxy\). The equation of the trajectory of point \(3^1\) in the plane \(Oxy\) has the form

\[ y = x \tg \varphi - b(\cos \varphi + \sin \varphi \tg \varphi). \]

The equation of the face surface that is formed during the driving of the borehole by the RUP-1 reamer is described by the system of equations:

\[ \begin{aligned} x &= (a + r \cos \theta)\cos \varphi + b \sin \varphi;\\ y &= (a + r \cos \theta)\sin \varphi - b \cos \varphi;\\ z &= -(V_{\text{feed}} t + r \sin \theta). \end{aligned} \]

The time during which the cutting edge of a tooth traverses the path \(S_0\) from the periphery of the face to the inner borehole is

\[ \tau_0 = \frac{\theta_k}{\omega_k}, \]

where \(\theta_k\) is the angular path of point \(3^1\) from the initial position \(3^1_0\) to the point corresponding to the tooth’s leaving contact with the face. Then the path \(S_0\) will be equal to:

\[ S_0 = \int_0^{\tau_0} v dt = \int_0^{\tau_0} \sqrt{ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2 }\cdot dt. \]

This expression cannot be integrated by means of functions known to us. In such cases one must abandon an exact analytical solution of the problem and seek an approximate solution.

Graphical-Analytical Solution of the Problem

Graphical-analytical methods are widely used in practice and, in particular, in studying the kinematics of the working members of mining machines [5]. They provide a visual solution of the problem and a sufficient degree of accuracy.

Figure 4 presents the distribution, over the face surface, of the working portions of the trajectories of the extreme points of the cutting edges of the teeth of crown I. By the working portion of a trajectory one must understand the path, during one revolution of the crown, traversed by the cutting edge of a tooth from the periphery to the inner borehole, along which the cutting process occurs.

The method of constructing the working portions of the trajectories is based on the following considerations:

1. The plane passing through the extreme points of the cutting edges of the teeth of the crown is the plane of their rotation. The trace of this plane in Fig. 3 for crown I is represented by the segment

$C_0D_0 = 2r$, where $r$ is the radius of rotation of the tooth edge in relative motion. During the translational rotation of the bit together with the body of the reamer about the borehole axis, the plane of rotation of the bit teeth throughout the entire motion occupies a position tangent to a cylinder of radius $r_0$, whose axis coincides with the borehole axis. We shall call this cylinder the basic cylinder, and its projection onto the plane $Oxy$ the basic circle. The radius of the basic circle is equal to the amount of displacement of the plane of rotation of the bit teeth relative to the parallel plane passing through the borehole axis (Fig. 3), i.e., $r_0 = A_0B_0$.

Fig. 4. Pattern of distribution over the face surface of the working sections of the trajectories of the extreme points of the cutting edges of the teeth of bit I.

1. The relative displacement of the extreme point of the cutting edge of a tooth takes place in the plane of rotation along a circle of radius $r$. When the plane of rotation of the bit is turned through an angle $\varphi$, together with the body of the reamer, about the borehole axis, the cutting edge of the tooth will describe an angular path $\Theta = 24.42\varphi$. As the initial position we shall take the point $3^1_0$ (a point on the periphery of the face). Then the displacement of point $3^1$ in the plane $Oxy$ from the periphery toward the inner borehole will be equal to the segment $3^1_0 m_i$, representing the projection of the displacement of point $3^1$ along the arc of a circle-

of radius \(r\) onto the trace of the plane of rotation of the bit (Fig. 3, b).

Analysis of the distribution pattern over the face surface of the working portions of the trajectories makes it possible to draw the following conclusions:

1. During one revolution of the reamer about the axis of the borehole \(\left(\tau = \dfrac{60}{n_p} = 15.54 \text{ sec.}\right)\), the total number of traces of the teeth of both bits on the face surface is \(q \approx 196\).

Fig. 5. Mutual arrangement of the portions of the tooth trajectories of the bits (a), the chip section during one revolution of the reamer (b), and the proposed scheme of chip removal (c):

1—inner borehole \(D_1\); 2—outer borehole \(D_2\); 3—working portion of the trajectory of tooth 2 of bit I; 4—working portion of the trajectory of tooth 1 of bit I; 5—working portion of the trajectory of a tooth of bit II.

1. The working portions of the trajectories of the teeth of bit I do not intersect with one another (Fig. 5, a). The distance between them at the inner borehole is \(\Delta S_1 = 12.55\) mm, and at the periphery, along the outer borehole, \(\Delta S_2 = 48.2\) mm.

1. The working sections have a slight curvature in plan, are close to straight lines, and therefore the tooth, as it moves along the face, selects (cuts off) a strip of coal of approximately constant width.

2. It is advisable to offset the teeth of the second cutter head relative to their position in Fig. 3, б by an angle \(\Delta \theta = 31.25^\circ\) in the direction of rotation of the cutter head, calculating it so that the working sections of the trajectories of its teeth pass between the working sections of the teeth of the first cutter head. This will increase the cutting surface and eliminate the repetition of the work of one tooth by another.

The mutual arrangement of the working sections of the trajectories of the teeth of the I and II cutter heads is shown in Fig. 5, а; in Fig. 5, б is shown the section of the chip along the length of the working stroke during 1 revolution of the reamer, and in Fig. 5, в the proposed chip-removal scheme. A “cellular” cutting scheme is adopted. \(\Delta H = 18\) mm is the depth of penetration of the tooth into the coal, equal to the feed of the reamer along the borehole axis per 1 revolution.

Fig. 6. Scheme of overlap of the working sections of the teeth of one cutter head.

1. Since the working sections of the trajectories in the \(Oxy\) plane have a slight curvature, and the feed along the borehole axis over the length of the working section of the tooth trajectory is small, the working length of the path \(S_0\) of one tooth during the time \(\tau_0 = \dfrac{\theta_k}{\omega_k} = 0.32\) sec can be determined, with sufficient accuracy, as the length of the circumference of radius \(r\) subtending the angle \(\theta_k = 180^\circ\). Then the total path traveled by all the teeth during one revolution of the reamer is equal to

\[ S = S_0 q = \frac{2\pi r}{360^\circ}\,\theta_k q \simeq 182\ \text{m}. \]

Fig. 6 shows a diagram of the overlap of the working sections of the teeth of one crown. Analysis of this picture makes it possible to draw a conclusion about the uniformity of loading of the crown during one revolution as it rotates about its axis. As the working section of the trajectory is traversed, the resultant velocity of the teeth changes only slightly and is of the order of \(v = 2.9\ \text{m/sec}\).

Calculation of Forces in Cutting Coal

The theory of coal cutting is at present still in such a state that it does not guarantee the obtaining of sufficiently accurate data when calculating the cutting forces of newly designed working members of mining machines. In order to establish the necessary theoretical relationships, numerous experimental studies and theoretical generalizations have been, and are being, carried out. As a result of these works, a number of empirical relationships have been proposed for determining the forces on the cutters of the most widespread mining machines and, in particular, of coal-cutting machines. In our opinion, the most valuable of the studies known to us for calculating the cutting force of coal by the working member of coal-cutting machines and bar-type combines are the works of M. M. Protodyakonov and S. S. Nekrasov [13—15].

M. M. Protodyakonov, in his investigations [13], came to the conclusion that, when calculating the forces of coal cutting by the cutters of coal-cutting machines, it is necessary to take into account the forces on the front and rear faces of the cutter. In the relationships derived by him, many factors manifesting themselves during cutter operation were taken into account, but for practical calculations they were very complicated. Thus, for example, it was recommended that the forces on the rear face be determined as a function of the contact area of the rear face of the tool with the coal. In practice, however, determining this area is very difficult. M. M. Protodyakonov related the forces on the front face to two strength characteristics: the compressive strength and the brittleness coefficient of the coal, which also made the calculations difficult. On the basis of his further investigations, M. M. Protodyakonov came to the conclusion that it was possible to simplify the calculation formulas and, in particular, for determining the average forces of coal cutting he proposed the formula

\[ P_{z\,\text{avg}} = 1.5\sigma_d \sqrt[3]{fL}, \]

where \(\sigma_d\) is the compressive strength of the coal;
\(f\) is the cross-sectional area of the cut;
\(L\) is the perimeter of the cut.

In the works of S. S. Nekrasov [15 and 16] published in recent years, data from the contact problem of the theory of elasticity are used to determine the principal component of the cutting force, and rather complicated calculation formulas are derived. However, in

in his final conclusions he arrives at empirical formulas of the power-law type. One of the shortcomings of these formulas is the variability of the exponents depending on

Fig. 7. Dependence of the cutting force \((P_{z\max}, \text{kg})\) on the cross-sectional area of the cut \((f, \text{mm}^2)\). Blocked cutting by a bit according to GOST 4617—49 \((\gamma = 14^\circ \cdot \alpha = 6^\circ)\); Donets coal \((\sigma_{d\max} = 5.6 \text{ kg/mm}^2)\):

\(\times\)—according to experiments; ○—according to Nekrasov’s equation; △—according to the authors’ equation.

the shape of the cutter. Thus, for example, for the case of blocked cutting by the bit of a cutting machine (GOST 4617—49) with angle \(\varphi = 0\)

Fig. 8. Dependence of the cutting force \((P_{z\max}, \text{kg})\) on the cross-sectional area of the cut \((f, \text{mm}^2)\). Blocked cutting by a bit according to GOST 4617—49 \((\gamma = 14^\circ \cdot \alpha = 6^\circ)\); Moscow-region coal \((\sigma_{d\max} = 2.5 \text{ kg/mm}^2)\):

\(\times\)—according to experiments; ○—according to Nekrasov’s equation; △—according to the authors’ equation.

the maximum cutting force is determined by the formula \(P_{z\max} \approx 6\sigma_{d\max} a^{0.87}\), while for bits of a somewhat different shape (GOST 4615—49 and 4616—49) \(P_{z\max} \approx 3.2\sigma_{d\max} a\), where \(a\) is the cutting depth. Hence it follows that, for calculating new cutters with another

geometry require different formulas, and new experimental studies are needed to derive them.

Moreover, if S. S. Nekrasov’s experimental data are compared with his theoretical formulas in ordinary coordinates, rather than in logarithmic coordinates as S. S. Nekrasov does, it turns out that these formulas do not always agree qualitatively or quantitatively with the experiment (Figs. 7–9). At the same time, in a related field of science, the theory of metal cutting, a large number of works and theoretical generalizations have been carried out, the results of which, in combination with experimental data on coal cutting, can be used in calculations of the working members of mining machines.

Fig. 9. Dependence of the cutting force \((P_{z\max}, \text{kg})\) on the cross-sectional area of the cut \((f, \text{mm}^2)\). Blocked cutting by a tooth according to GOST 4615—49 \((\gamma = 20^\circ;\ \alpha = 25^\circ)\); under-cutter angle \((\sigma_{d\max} = 2.5\ \text{kg}/\text{mm}^2)\); ×—from experiments; ○—according to Nekrasov’s equation; △—according to the authors’ equation.

In recent years, in the laboratory of the Department of Metal Cutting of the Tomsk Polytechnic Institute, on the basis of extensive experimental studies, the basic principles of the theory of cast-iron cutting have been developed [6—11]. For calculating the cutting forces of cast iron, relations are given that take into account the presence of forces on the front and rear faces of the cutting element and their connection with the parameters of the cutting process:

\[ P_z = A_1 \sigma_d f + B_1 \sigma_d L, \tag{1} \]

\[ R_{xy} = A_2 \sigma_d f + B_2 \sigma_d L, \tag{2} \]

where \(P_z\) is the vertical component of the total cutting force;
\(R_{xy}\)—the horizontal component of the total cutting force;
\(\sigma_d\)—the compressive strength;
\(f\)—the cross-sectional area of the cut;
\(L\)—the length of the cut perimeter;
\(A_1\) and \(A_2\)—proportionality coefficients depending on the leading angle and cutting temperature;

\(B_1\) and \(B_2\) are proportionality coefficients that depend on the clearance angle and the cutting temperature.

If we assume that the basic propositions of the theory of cutting gray cast iron also apply to coal cutting, then these relationships can be used for the calculations we require. In this case it should be expected that the forces on the front and rear faces of the cutting element will depend to a lesser degree on the cutting temperature (or on the cutting speed), and, consequently, the proportionality coefficients \(A_1\), \(A_2\), \(B_1\), \(B_2\) will depend mainly on the geometry of the tool.

When calculating \(P_{z \max}\), we take \(\delta_{d\max}\), and for the case of blocked cutting we shall have:

\[ f = ab_0 + a^2 \operatorname{tg}\xi, \]

\[ L = b_0 + 2 \frac{a}{\cos \xi}. \]

Here \(b_0\) is the width of the cutting edge of the cutter in the narrow part;
\(a\) is the width of cut;
\(\xi\) is the angle in plan (the included angle).

Using the experimental data of Nekrasov, we determined the coefficients \(A_1\) and \(B_1\) in equation (1). When cutting Donets and Moscow-region coals with teeth (GOST 4617—49), the values of both \(A_1\) and \(B_1\) coincide quite closely. For both coals they may be taken as \(A_1 = 0.32\) and \(B_1 = 0.6\). The teeth according to GOST 4617—49 have the following geometry: rake angle \(\gamma = 14^\circ\), clearance angle \(\alpha = 6^\circ\), width in the lower (rear) part \(b_0 = 7.5\) mm. For teeth according to GOST 4615—49 (\(\gamma = 20^\circ\), \(\alpha = 25^\circ\), \(b_0 = 3.5\) mm), the coefficient values are \(A_1 = 0.3\) and \(B_1 = 0.25\). As was to be expected, the slight decrease in coefficient \(A_1\) in this case is caused by an increase in the rake angle, while the considerable decrease in coefficient \(B_1\) is caused by the large increase in the cutter clearance angle. Similar calculations carried out for different chip sections gave results that agree well with the experimental data both of S. S. Nekrasov (Figs. 7—9) and of other investigators.

When calculating \(P_{y \max}\) for the case of blocked cutting,

\[ L' = b_0 + 2a \operatorname{tg}\xi . \]

For cutting Donets and Moscow-region coals with teeth according to GOST 4617—49, the values of the proportionality coefficients are \(A_2 = 0.2\) and \(B_2 = 0.3\). From Fig. 10 one can see good qualitative and quantitative agreement of these calculated data as well with the experiment.

On the basis of the foregoing, we consider it quite possible, for calculating the forces of coal cutting by the cutters (teeth) of the RUP-1 reamer, to use the theoretical relationships,

derived for cutting gray cast iron, together with experimental data on cutting coal [13—15].

On the basis of studies of coal cutters of various shapes [15—16], for calculating the cutting forces on the cutters of the RUP-1 reamer we adopt the following geometry: \(\gamma = 15^\circ\), \(\alpha = 15^\circ\), \(\alpha_1 = 15^\circ\), and \(\xi = 15^\circ\). To ensure the lowest energy consumption, the ratio between the chip thickness \(a\) and the chip width \(b\) must be

Fig. 10. Dependence of the cutting force \((P_{y\max}, \text{kg})\) on the cross-sectional area of the chip \((f, \text{mm}^2)\):
\(\Delta\)—according to Nekrasov; \(\times\)—experimental data; \(\circ\)—according to the authors’ equation;
\(1\)—\(\sigma_{d\max} = 5.6\ \text{kg/mm}^2\);
\(2\)—\(\sigma_{d\max} = 2.5\ \text{kg/mm}^2\).

\[ a > \frac{b}{2} \]

[16]. Taking into account the adopted cutting scheme (Fig. 5), we adopt a blade-shaped cutter with \(b_0 = 12\ \text{mm}\). According to V. N. Solod [16], at a pitch \(t = (2.5 \div 3)a\), blocked (three-sided) cutting takes place, and at \(t \approx a\), semi-blocked (two-sided) cutting takes place. In our case the pitch varies from \(t = 48.2\ \text{mm}\) to \(t = 12\ \text{mm}\), while the chip width at the initial moment is equal to zero, then reaches a maximum \((a_{\max} = 9\ \text{mm})\), and subsequently again decreases to zero. On this basis, it may be considered that the RUP-1 cutters perform blocked cutting.

Since the motion of the cutter of the reamer relative to the instantaneous axis of rotation may be regarded as motion occurring along a helical line with a very small angle of inclination \((\alpha_{\max} \approx 3^\circ 30' \div 4^\circ)\) at the moment of cutting, and \(\alpha = 0\) at \(\alpha_{\max}\), it may be considered that the cutter will be acted upon mainly by two forces, \(P_z\) and \(P_y\). We calculate the maximum values of these forces on the sharp cutter from the equations:

\[ P_{z\max}=A_1\sigma_{d\max} f_\varphi + B_1\sigma_{d\max} L_\varphi, \tag{3} \]

\[ P_{y\max}=A_2\sigma_{d\max} f_\varphi + B_2\sigma_{d\max} L'_\varphi. \tag{4} \]

For coal of the Kuznetsk Basin, \(\sigma_{d\max}=3\ \mathrm{kg/mm^2}\).

On the basis of the analysis given above, for the specified tool geometry \(\gamma=15^\circ\) and \(a=15^\circ\), we take \(A_1=0.32\), \(B_1=0.6\), \(A_2=0.2\), \(B_2=0.3\). The chip thickness depends on the angle of rotation of the reamer \(\varphi\) [12]

\[ a_\varphi=t_1\sin\varphi, \tag{5} \]

where \(t_1\) is the depth of cutting. Then we have:

\[ f_\varphi=t_1 b_0\sin\varphi + t_1^2\sin\varphi\,\tan\xi, \tag{6} \]

\[ L_\varphi=b_0+2\frac{t_1\sin\varphi}{\cos\xi}, \tag{7} \]

\[ L'_\varphi=b_0+2t_1\sin\varphi\,\tan\xi. \tag{8} \]

The force \(P_{z\max}\) will create the maximum torque on the reamer

\[ M_{\mathrm{torq.}\max}=P_{z\max}r, \tag{9} \]

where \(r\) is the radius of the reamer.

The force \(P_{y\max}\) will create pressure on the axis of the reamer and the maximum feed force

\[ P_{\mathrm{feed.}\max}=P_{y\max}\sin\varphi. \tag{10} \]

Having determined the values of \(P_{\mathrm{feed.}\max}\) and \(M_{\mathrm{torq.}\max}\), we construct the dependences of these quantities on \(\varphi\) (Figs. 11, 12). Since two cutters of the reamer will simultaneously be under the chip, we determine the total torque and the total feed force. As the cutters wear, the cutting force will increase. According to S. S. Nekrasov [14], at optimum wear on the clearance face \(\Delta=3\ \mathrm{mm}\), the force \(P_{z\max}\) increases twofold, and \(P_{y\max}\) one and a half times. As a result of the calculation we have:

\[ \text{total } M_{\mathrm{torq.}\max\,\Delta=3\ \mathrm{mm}}=150\ \mathrm{kgm}, \]

\[ \text{total } P_{\mathrm{feed.}\max\,\Delta=3\ \mathrm{mm}}=142\ \mathrm{kg}. \]

The maximum values of torque and forces are required in calculating the machine for strength. To determine the required

Figure 11. Dependence of the magnitude of the torque \((M_{\mathrm{torque}\,\max}, \text{kg·m})\) on the angle \((\varphi^\circ)\) of rotation of the cutter.

Labels in the figure:

• Vertical axis: Maximum torque magnitude, kg·m
• Horizontal axis: Angle of rotation of the cutter, degrees
• \(M_{\mathrm{torque}}\), total
• \(M_{\mathrm{torque}}\) on tooth 4
• \(M_{\mathrm{torque}}\) on tooth 1
• \(M_{\mathrm{torque}}\) on tooth 2

Figure 12. Dependence of the maximum value of the feed force \((P_{\mathrm{feed}}, \text{kg})\) on the angle \((\varphi^\circ)\) of rotation of the cutter.

Labels in the figure:

• Vertical axis: Maximum feed-force magnitude, kg
• Horizontal axis: Angle of rotation of the cutter, degrees
• \(P_{\mathrm{feed}}\), total
• \(P_{\mathrm{feed}}\) on tooth 4
• \(P_{\mathrm{feed}}\) on tooth 1
• \(P_{\mathrm{feed}}\) on tooth 2

to determine the power it is necessary to know the average values of these quantities. According to S. S. Nekrasov and M. M. Protodyakonov [13, 15]

\[ P_{z\,\mathrm{avg}} \approx 0.5 P_{z\,\max}, \tag{11} \]

\[ P_{y\,\mathrm{avg}} \approx 0.7 P_{y\,\max}. \tag{12} \]

Therefore:

\[ \text{total } M_{\mathrm{cr.\,avg}\,\Delta=3\,\mathrm{mm}} = 75\ \mathrm{kgm}, \]

\[ \text{total } P_{\mathrm{feed.\,avg}\,\Delta=3\,\mathrm{mm}} = 100\ \mathrm{kg}. \]

Because the cutter of the reamer moves along a helical line, owing to the deviation of the friction force a force \(P_x\) will arise, which creates a torque about the axis of the borehole. According to approximate calculations \((P_x = P_y \tan \alpha)\), where \(\alpha\) is the angle of inclination of the helical line, \(M'_{\mathrm{cr.\,max}} = 1\ \mathrm{kgm}\).

In conclusion it is necessary to note the following: in calculating the forces, the influence of the cutting speed has not been taken into account, nor has the pressure of the broken coal on the cutters of the reamer. The entire calculation has been carried out for one cutterhead (crown), having four cutters (teeth).

Determination of the total moment on the driving shaft

As was stated above, the reamer is driven by an SBM-3u machine, connected with it by means of a drill string. Let us denote:

\(M\) — the torque on driving shaft 15 (Fig. 1), representing the external moment transmitted through the drill string from the shaft of the SBM-3u motor;

\(M_{\mathrm{cor.\,1}}\) — the torque on wheel 1, used to overcome the moments of resistance associated with the operation of cutterhead I or II;

\(M_5 = 2\ \mathrm{kgm}\) — the torque on wheel 5, equal to the moment of resistance arising when the reamer rotates about the axis of the borehole; we take it as equal to twice the average moment of resistance to cutting, \(2M'_{\mathrm{cr.\,max}}\), when the cutterhead moves around the axis of the borehole.

The torque on the driving shaft is expended in overcoming the moments of resistance associated with the operation of two cutterheads and with the rotation of the reamer; consequently

\[ M = 2M_{\mathrm{cor.\,1}} + M_5. \]

A study of the statics of the gear transmission from wheel 1 to cutterhead I makes it possible to establish the following relation:

\[ M_{\mathrm{cor.\,1}} = \frac{M_k}{i\eta_{12}\eta_{34}}, \]

where \(M_k\) is the moment of resistance to cutting at cutterhead I (or II). We take:

\[ M_k = M_{\mathrm{cr.\,avg}\,\Delta=3\,\mathrm{mm}} = 75\ \mathrm{kgm}; \]

\[ i=\frac{z_2}{z_1}\cdot\frac{z_4}{z_3}=\frac{1}{2} \]

— the gear ratio from shaft 15 to the crown shaft;

\[ \eta_{12}=0.95 \]

— efficiency of gear transmission 1—2;

\[ \eta_{34}=0.95 \]

— efficiency of gear transmission 3—4;

Then the total torque on the driving shaft 15 will be \(M=334\ \text{kgm}\).

Determination of the Power Required for Operation of the Reamer

The power supplied through the drill string to shaft 15 of the reamer is distributed in the following four directions:

1) \(N_1\) — the power required to overcome the cutting resistance at crown I;

2) \(N_2\) — the power required to overcome the cutting resistance at crown II;

3) \(N_3\) — the power expended in rotating the reamer about the axis of the borehole;

4) \(N_4\) — the power expended in overcoming the resistances associated with feeding the reamer along the axis of the borehole,

\[ N_1+N_2+N_3=0.0010267Mn_1=17.45\ \text{kW}. \]

The power \(N_4\) consists of:

a) the power \(N'_4\), used to overcome the work of the feed-resistance force \(P_{\text{feed}}\) of cutting on the teeth of the crowns along the axis of the borehole;

b) the power \(N''_4\), expended in overcoming the resistance forces arising at cutters 12 (Fig. 1) of the brake lantern when the reamer moves along the axis of the borehole:

\[ N_4=N'_4+N''_4, \]

\[ N'_4=\frac{P_{\text{feed}}V_{\text{feed}}}{102}=0.002\ \text{kW}, \]

where \(P_{\text{feed}}=200\ \text{kg}\) is the feed-resistance force on both crowns.

The resistance force at cutters 12 is equal to

\[ F=2af\ [6], \]

where \(a\) is the coefficient of resistance of coal to cutter motion;
\(f\) is the area of the cutter.

Then

\[ N''_4=\frac{2afV_{\text{feed}}}{102}\ \text{kW}. \]

Since the value of \(a\) is unknown to us, we take \(N''_4\approx N'_4\).

The power required for operation of RUP-1 is

\[ N=N_1+N_2+N_3+N_4\approx17.5\ \text{kW}. \]

The study carried out showed that the design of the RUP-1 reamer can provide for the enlargement of coal chute holes from a diameter of 300 to 1500 mm when the SBM-3u drilling-and-breaking machine is used as the drive.

REFERENCES

1. O. D. Alimov, D. N. Malyakov, I. A. Ushakov. On the question of driving raises in coal mines of the Prokopievsk-Kiselevsk district of Kuzbass. Article in the present collection.

2. SBM-3u drilling-and-breaking machine. Operating and maintenance manual. Ugletekhizdat, 1948.

3. N. F. Rudenko. Planetary transmissions. Mashgiz, 1947.

4. L. B. Levenmon. Theory of Mechanisms and Machines. Mashgiz, 1954.

5. G. I. Mankovskii, A. N. Vladimirov. Drilling of Shaft Bores by Means of Chain Reamers. Ugletekhizdat, 1950.

6. Yu. A. Rozenberg. Investigation of the Process of Cutting Gray Cast Iron. Candidate’s dissertation. Library of TPI, Tomsk, 1952.

7. A. M. Rozenberg, Yu. A. Rozenberg. The influence of deformation rate and temperature on stress in cutting brittle metal. Izvestiya TPI, vol. 75, Tomsk, 1954.

8. A. M. Rozenberg, Yu. A. Rozenberg. Equation of the force in cutting brittle metal (cast iron). Izvestiya TPI, vol. 75, Tomsk, 1954.

9. A. K. Baikalov. Optimal Geometry of Cutters in High-Speed Turning of Gray Cast Iron. Candidate’s dissertation. Library of TPI, Tomsk, 1953.

10. L. M. Sedokov. Dynamics of Face Milling of Gray Cast Iron. Candidate’s dissertation. Library of TPI, Tomsk, 1956.

11. A. M. Rozenberg, A. N. Eremin. Elements of the Theory of the Metal-Cutting Process. Mashgiz, 1956.

12. A. M. Rozenberg. Dynamics of Milling. Sovetskaya Nauka, 1945.

13. M. M. Protodyakonov. Theory of Coal Cutting by Chain Cutter-Loader Machines. ONTI NKTP USSR, 1936.

14. S. S. Nekrasov. Investigation of forces in the destruction of coal by cutting. Ugol’, No. 11, 1954.

15. S. S. Nekrasov. Elements of the theory of resistance of brittle materials to cutting. Collection of Scientific Works of MGI, No. 15, Moscow, 1955.

16. V. N. Solod. Some questions of the efficiency of coal destruction by the working members of machines operating on the principle of large chipping. Candidate’s dissertation. Library of MGI, Moscow, 1954.

NOTED MISPRINTS

Order 39

1. O. D. Alimov, V. N. Karminskii, D. N. Malikov. Reamer for coal chutes RUP-1. Application No. 561065/22. Committee for Inventions and Discoveries under the Council of Ministers of the USSR. ↩