Effects of ranging arm housing’s large deformation on the dynamic responses of the cutting transmission system for a drum shearer

Abstract

The drum driving system is the weakest part of a drum shearer, and most of the previous dynamic analyses neglect the housing flexibility. Therefore, some efforts are needed to refine the dynamic model and sufficiently understand the dynamic behavior of the drum driving system. Establishing a system-level model of the drum driving system is challenging because the drum driving system consists of many components such as the cantilever loaded housing, multistage gear transmission, electric motor, and drum. In this study, an improved three-dimensional gear model with the time-varying pressure angle and contact ratio due to center distance variation is proposed. Then, a standard coupling method is proposed to construct the hybrid dynamic model of the drum driving system automatically. Finally, a comparative analysis of the dynamic responses of the drum driving system under two loading scenarios (i.e. applying only the cutting moment and applying both the cutting moment and cutting force) revealed that the three-directional cutting force was the primary external factor that leads to the housing deformation. And some principles for designing the drum driving system under such a large deformation condition are also identified.

Keywords

Drum shearer housing deformation transmission system gear dynamics substructure dynamic model

Introduction

The drum shearer is a key equipment in the long-wall mining system, which has been the dominant coal mining method for decades in the world. The stable and reliable operation of the drum shearer is highly critical to safe and efficient production on a completely mechanized coal mining face.¹ The drum driving system is the pivotal component for realizing coal cutting and consumes 80%–90% of the total power used by the shearer.² Figure 2 shows a typical schematic of the drum driving system of a shearer, which consists of an asynchronous motor, a gearing system, a drum, and a ranging arm housing. According to the recent statistics,³ the drum driving system is the most vulnerable component of the shearer. As illustrated in Figure 1, the ranging arm housing is cantilevered and prone to result in a large deformation by the fluctuant drum load. Housing’s large deformation will make the dynamic characteristics of the transmission system different from the ideal state. Therefore, the impacts of a large housing deformation on the dynamic characteristics of the transmission system should be taken into consideration at an early design stage, and some principles for designing the drum driving system with a large housing deformation should be proposed.

Figure 1.

Drum shearer and its working principle: (a) photograph of the actual drum shearer and (b) schematic illustrating the working principle.

As an electromechanical system with multistage gears, the systematic dynamic response of the shearer’s drum driving system has received considerable attention from researchers and engineers. Dolipski et al.^4–6 described a lumped-parameter dynamic model of a shearer’s cutting system and verified it through experimental tests, and the interactions of coal hardness and hauling speed on gear meshing force were also analyzed. Jiang et al.⁷ established a nonlinear gear dynamic model of the drum driving system with multi-frequency excitation load and developed a useful indicator based on maximal Lyapunov exponent to identify the gear crack degrees. Liu et al.⁸ proposed an electromechanical dynamic model including the electric motor and the electromechanical dynamic characteristics of the drum driving system were simulated during the speed change process. Ge et al.⁹ established a torsional electromechanical coupling model of a drum driving system and investigated the control strategy for suppressing the dynamic load of the gear transmission system under impact load. However, the coupling effects of the ranging arm housing deformation and the transmission system were neglected in the above-mentioned studies. Previous studies show that the nucleus of the gear system dynamic model is the modeling of gear tooth contact actions.¹⁰,¹¹ Since housing deformation will indeed change the instant screw axes of a pair of gears and then change the tooth contact action,^12–14 the housing flexibility should be included in the prediction process of dynamic responses. Through the virtual prototyping of a drum driving system using the ADAMS software, Zhao and Ma¹⁵ calibrated the stress of the ranging arm housing and deformation of the planet carrier. However, because of the simplified gear dynamic model, this study did not consider the effects of the deformation of ranging arm housing on the engaging state of the gears. Zhao and Lan¹⁶ modeled one pair of gears through finite element method in their virtual prototype of a ranging arm to solve contact problems and obtained the contact stress on the tooth flanks under a steady cutting load. However, this model focused only on one pair of gears among a large number of gears and failed to investigate the overall dynamic characteristics of the complete drum driving system under the ranging arm housing deformation conditions. Qin and Jia¹⁷ established a dynamic model of the complete drum driving system using the hybrid dynamic substructure method that combined the finite element model of housing and the lumped-parameter model of the transmission system. Their model was validated qualitatively through experiments, and then the dynamic meshing force and dynamic equivalent mesh misalignment under housing deformation were calculated through computer simulation. However, the gear dynamic model that Qin and Jia¹⁷ used to compose the drum driving system was not close enough to reality, since it neglected the fluctuation of pressure angle and contact ratio due to the gear center distance variation under housing deformation conditions. Recent research^18–20 indicated that the variation of gear center distance will change the pressure angle and contact ratio, which observably influence the gear dynamic responses. So the pressure angle and contact ratio should be regarded as the time-varying variables in the hybrid dynamic model of the drum driving system. Since these studies^18–20 restricted the motion of the gears within a two-dimensional (2D) plane, a new three-dimensional (3D) parallel-axis gear model should be proposed considering the time-varying pressure angle and contact ratio.

In this study, considering the gear center distance variation, a 3D parallel-axis gear model with time-varying pressure angle and contact ratio is proposed for the first time, which can be viewed as a generalization of the previous studies.^18–20 Then, the gear dynamic model in the authors’ previous work¹⁷ is replaced by the newly proposed more realistic one, and other subcomponent models are inherited directly, that is, housing model, planetary gear model, gear shaft model, asynchronous motor model, and random drum load model. So a better theoretical accuracy of the hybrid dynamic model can be expected. In addition, a standard and automatic coupling method using matrix calculation is proposed to replace the previous manual matrix transformations.¹⁷ That means the proposed coupling method can improve the speed of modeling and reduce the possibility of error at the same time. A comparative analysis of the dynamic characteristics of the drum driving system under two loading scenarios, that is, M (applying only the cutting moment) and M + F (applying both cutting moment and cutting force), showed that the three-directional cutting force was the primary external factor leading to the large deformation of the ranging arm housing. Then, the impacts of this large deformation on the dynamic characteristics of the key components of the transmission system (such as gears, bearings, and flexible shaft) were examined. The analysis results could provide guidance in the design and simulation of a drum driving system under a large housing deformation condition.

Shearer and drum driving system

Figure 1(a) illustrates an actual drum shearer. The guide rail supports and guides the shearer body. The hydraulic cylinder regulates the angle of the ranging arm, which allows the drum to follow the height of the coal seam. The cutting picks around the periphery of a drum produce the coal cutting action when the drum rotates. During operation, the shearer advances at a certain hauling speed on the guide rail, while the front (upper) and rear (lower) drums operate in a coordinated manner, which makes it possible to cut the entire height of the mine face in one run. Figure 2 shows a schematic of the drum driving system of a 1.2 MW shearer, which consists of an asynchronous motor, a gearing system, a drum, and a ranging arm housing. The gearing system includes two-stage parallel-axis gears (Z₁₁–Z₁₃ and Z₂₁–Z₂₆) and two-stage planetary gears. The ranging arm housing is manufactured using ZG25Mn2 through integral casting, with a Young modulus of 202 GPa and a Poisson ratio of 0.3.

Figure 2.

Schematic of the drum driving system of a 1.2 MW shearer.

Dynamic modeling of the drum driving system

An improved 3D gear dynamic model

In this study, the classical gear dynamic model, the pseudo-3D thin-slice approach²¹ is improved by regarding pressure angle and contact ratio as the time-varying variables. The pinion and gear are assimilated to two rigid cylinders with 6 degrees of freedom (DOFs) each, and the generalized displacements of gear $g_{j}$ are denoted as $u^{gj} = [x^{gj}, y^{gj}, z^{gj}, θ_{x}^{gj}, θ_{y}^{gj}, θ_{z}^{gj}]^{T}$ . Then it is assumed that the actual gear pairs can be viewed as the superposition of infinitesimal mesh elements in the face width direction. Each of these mesh elements has the time-varying pressure angle and contact ratio. As shown in Figure 3, the dynamic equation of this gear pair can be expressed in the matrix form as follows

[\begin{matrix} M^{g 1} \\ M^{g 2} \end{matrix}] {\begin{matrix} {\overset{\cdot\cdot}{u}}^{g 1} \\ {\overset{\cdot\cdot}{u}}^{g 2} \end{matrix}} = {\begin{matrix} R^{g 1} \\ R^{g 2} \end{matrix}}

(1)

where $M^{gj} = diag (m^{gj}, m^{gj}, m^{gj}, I_{x}^{gj}, I_{y}^{gj}, I_{z}^{gj}), j = 1, 2$ ; “diag” denotes a diagonal matrix; $m^{gj}$ is the mass of gear $g_{j}$ ; $I_{k}^{gj} (k = x, y, z)$ is the moment of inertia about the k-axis at the gear center; and $R^{gj} = [F^{gj}, T^{gj}]^{T}, j = 1, 2$ , is the total dynamic meshing load of gear $g_{j}$ , which consists of dynamic meshing force $F^{gj}$ and dynamic meshing moment $T^{gj}$ .

Figure 3.

Parallel-axis gear model.

As illustrated in Figure 3, the gear center is denoted as $g_{j}$ and the centers of infinitesimal mesh elements of gear $g_{j}$ are denoted as $c_{i}^{gj} (j = 1, 2; i = 1, 2, 3, \dots)$ . The total dynamic meshing force and meshing moment of gear $g_{j}$ are integrals of the micro-loads on these infinitesimal mesh elements

F^{gj} = \int_{b} d F^{gj} = \int_{b} - (c_{e} \overset{\cdot}{δ} (M^{i}) + k_{e} δ (M^{i})) \cdot n_{i}^{gj}

(2)

T^{gj} = \int_{b} d T^{gj} = \int_{b} - (c_{e} \overset{\cdot}{δ} (M^{i}) + k_{e} δ (M^{i})) \cdot r_{i}^{gj} \times n_{i}^{gj}

(3)

where $b$ is the tooth width; $d F^{gj}$ is the 3D dynamic meshing force vector at an infinitesimal mesh element; $d T^{gj}$ is the 3D dynamic meshing moment vector of $d F^{gj}$ with respect to the center of gear $g_{j}$ ; $c_{e}$ and $k_{e}$ are the damping and stiffness of each mesh element, respectively; $δ (M^{i})$ and $\overset{\cdot}{δ} (M^{i})$ are the normal deviation and its derivative at any potential contact point $M^{i}$ , respectively; $n_{i}^{gj}$ is the outward unit normal vector of infinitesimal mesh element $c_{i}^{gj}$ on the mesh plane; $r_{i}^{gj}$ is a 3D vector pointing from the gear center $g_{j}$ to the potential contact point $M^{i}$ ; and × means the cross product of two vectors.

The core of equations (2) and (3) is calculating $d F^{gj}$ and $d T^{gj}$ , which are the micro-loads on the infinitesimal mesh element. The translational motion $((x^{gj}, y^{gj}, z^{gj}), j = 1, 2)$ and axil swing $((θ_{x}^{gj}, θ_{y}^{gj}), j = 1, 2)$ of the gears lead to changes in the center distances of infinitesimal mesh elements and thus also in the pressure angles and contact ratios of the elements. Two steps are needed to calculate the micro-loads: (1) determining the generalized displacements of the infinitesimal gear center by equations (4) and (5), and (2) determining the normal deviation and its derivative of the infinitesimal mesh element by equations (6) and (7).

The generalized displacements of infinitesimal gear center $c_{i}^{gj}$ are expressed as follows

x^{gji} = x^{gj} + b_{i} \cdot θ_{y}^{gj}

(4)

y^{gji} = y^{gj} - b_{i} \cdot θ_{x}^{gj}

(5)

where $x^{gj}$ , $y^{gj}$ , $θ_{x}^{gj}$ , and $θ_{y}^{gj}$ are the generalized displacements of gear center $g_{j}$ and $b_{i}$ is the z-axis coordinate of $c_{i}^{gj}$ .

Following del Rincon et al.,²⁰ the normal deviation and its derivative of the infinitesimal mesh element are expressed as follows

\begin{matrix} δ (M^{i}) & = (r_{b 1} + r_{b 2}) (\tan (α) + α_{v} - α \mp Ψ) \\ \pm (r_{b 1} θ_{z}^{g 1} + r_{b 2} θ_{z}^{g 2}) - d_{v} \sin (α_{v}) \end{matrix}

(6)

\begin{matrix} \overset{\cdot}{δ} (M^{i}) & = (r_{b 1} + r_{b 2}) ({\overset{\cdot}{α}}_{v} \mp \overset{\cdot}{Ψ}) \\ \pm (r_{b 1} {\overset{\cdot}{θ}}_{z}^{g 1} + r_{b 2} {\overset{\cdot}{θ}}_{z}^{g 2}) - d_{v} \sin (α_{v}) \end{matrix}

(7)

where $r_{bj} (j = 1, 2)$ is the base circle radius; $α$ is the nominal pressure angle; $α_{v}$ is the time-varying pressure angle obtained by equation (10); $Ψ$ represents the position of the gear relative to the pinion, which is obtained by equation (8); and $d_{v}$ is the center distance between $c_{i}^{g 1}$ and $c_{i}^{g 2}$ obtained by equation (9). It is worth noting that the upper sign is for pinion rotating anti-clockwise which is shown in Figure 3, while the lower sign is for pinion rotating clockwise

Ψ = \arctan (\frac{y^{g 2 i} - y^{g 1 i}}{d + x^{g 2 i} - x^{g 1 i}})

(8)

d_{v} = \sqrt{{(d + x^{g 2 i} - x^{g 1 i})}^{2} + {(y^{g 2 i} - y^{g 1 i})}^{2}}

(9)

α_{v} = \arccos (\frac{r_{b 1} + r_{b 2}}{d_{v}})

(10)

ε_{v} = \frac{1}{2 π} [z_{1} (\tan (α_{a 1}) - \tan (α_{v})) + z_{2} (\tan (α_{a 2}) - \tan (α_{v}))]

(11)

where $ε_{v}$ is the time-varying contact ratio; $z_{j} (j = 1, 2)$ is the tooth number for each gear; and $α_{aj} (j = 1, 2)$ is the addendum circle pressure angle of gear $g_{j}$ . Since $\overset{\cdot}{Ψ}$ , ${\overset{\cdot}{d}}_{v}$ , and ${\overset{\cdot}{α}}_{v}$ can be derived from equations (8)–(10) easily, they are not presented in this article (Figure 4).

Figure 4.

The model of infinitesimal mesh element (a) at ideal position and (b) with nonstandard center distance.

Subcomponent dynamic model in the matrix form

The dynamic models of ranging arm housing, gear shafts, planetary gears, asynchronous motor, and drum load have been established by the authors in their previous work,¹⁷ which will be used directly in this work. These subcomponent models and the parallel-axis gear model proposed in the above section will be expressed in the unified matrix terms. The benefit is that researchers and engineers can couple different subcomponent models together automatically through matrix calculations, especially when the transmission system consists of many subcomponents

\begin{matrix} [M^{i}] {{\overset{\cdot\cdot}{u}}^{i}} + [C^{i}] {{\overset{\cdot}{u}}^{i}} + [K^{i}] {u^{i}} \\ = {r^{i} (t)} + {R^{i} ({\overset{\cdot}{u}}^{i}, u^{i}, t)} (i = h, g, p, s, m, d) \end{matrix}

(12)

where the superscripts denote different subcomponents: ranging arm housing $(h)$ , parallel-axis gears $(g)$ , planetary gears $(p)$ , gear shafts $(s)$ , asynchronous motor $(m)$ , and drum $(d)$ ; $M^{i}$ , $C^{i}$ , and $K^{i}$ are the mass matrix, damping matrix, and stiffness matrix, respectively, and they represent the linear functions of a dynamic model; $R^{i} ({\overset{\cdot}{u}}^{i}, u^{i}, t)$ represents the nonlinear functions of a dynamic model; $r^{i} (t)$ is the external load and can be time-varying or constant.

A standard coupling method

As illustrated in Figure 5, the drum driving system consists of the ranging arm housing, parallel-axis gears, planetary gears, gear shafts, asynchronous motor, and the drum. These subcomponents are connected through three kinds of connections: linear flexible connection, nonlinear flexible connection, and rigid connection. Since the dynamic models of subcomponents are established under the unified coordinate, no coordinate transformation is needed when coupling them together. Therefore, only two steps are needed to construct the coupling dynamic model of the drum driving system:

Step 1. Stack the subcomponent models up directly in the form of block diagonal matrix

[M] {\overset{\cdot\cdot}{u}} + [C] {\overset{\cdot}{u}} + [K] {u} = {r (t)}

(13)

where $u = [u^{h}, u^{g}, u^{p}, u^{s}, u^{m}, u^{d}]^{T}$ represents the generalized displacements of all the subcomponents; $M$ , $C$ , and $K$ are the block diagonal matrices, that is, $M = diag (M^{h}, M^{g}, M^{p}, M^{s}, M^{m}, M^{d})$ .

As shown in Figure 5, the node numbers of boundary points on the housing range from 1 to 17, the node numbers of parallel-axis gears are from 18 to 26, the node numbers of planetary gears are from 27 to 37, the node numbers of gear shafts are from 38 to 77, the node number of motor is 78, and the node number of drum is 79. At this step, there are 79 nodes in total.

Step 2. Obtain the coupling dynamic model through matrix calculation

Equation (13) has not considered the three kinds of connections illustrated in Figure 5. This step demonstrates how to make the independent subcomponent models coupled through matrix calculation. Since two rigid-connected nodes have identical displacement, speed, and acceleration, they can be viewed as a single node. As shown in Figure 5, 14 nodes can be eliminated by 14 rigid connections. Therefore, the coupling drum driving system has 65 nodes in total finally. The matrix calculations for the three kinds of connections are demonstrated as follows:

Matrix calculation for nonlinear flexible connection

After adding nonlinear flexible connections to these subcomponent models, equation (13) can be rewritten as follows

[M] {\overset{\cdot\cdot}{u}} + [C] {\overset{\cdot}{u}} + [K] {u} = {r} + {R^{Non} (\overset{\cdot}{u}, u, t)}

(14)

Without loss of generality, take such a system as an example: four nodes in total, node 1 and node 3 are coupled through nonlinear flexible connection. The calculation of $R^{Non} (\overset{\cdot}{u}, u, t)$ is demonstrated as follows.

Define

N^{Non} = [\begin{matrix} I & 0 \\ 0 & 0 \\ 0 & I \\ 0 & 0 \end{matrix}]

(15)

Therefore

\begin{matrix} {R^{Non} (\overset{\cdot}{u}, u, t)} & = {\begin{matrix} R_{1} ({[{\overset{\cdot}{u}}_{1}, {\overset{\cdot}{u}}_{3}]}^{T}, {[u_{1}, u_{3}]}^{T}, t) \\ 0 \\ R_{3} ({[{\overset{\cdot}{u}}_{1}, {\overset{\cdot}{u}}_{3}]}^{T}, {[u_{1}, u_{3}]}^{T}, t) \\ 0 \end{matrix}} \\ = N^{Non} * {\begin{matrix} R_{1} ({[N^{Non}]}^{T} \overset{\cdot}{u}, {[N^{Non}]}^{T} u, t) \\ R_{3} ({[N^{Non}]}^{T} \overset{\cdot}{u}, {[N^{Non}]}^{T} u, t) \end{matrix}} \end{matrix}

(16)

where $I$ is an identity matrix of order 6; $0$ is a zero matrix of order 6; $R_{1}$ and $R_{3}$ are the known nonlinear functions of ${\overset{\cdot}{u}}_{1}$ , ${\overset{\cdot}{u}}_{3}$ , $u_{1}$ , $u_{3}$ , and $t$ , among which $t$ is the time variable; since $[N^{Non}]^{T} u = [u_{1}, u_{3}]^{T}$ , $R^{Non}$ can be written as a nonlinear function of $\overset{\cdot}{u}$ and $u$ , and $t$ through coordinate transformation.

If the number of nonlinear flexible connections is greater than one, add these connections one by one

R^{Non} = R^{Non_1} + R^{Non_2} + \dots + R^{Non_N}

(17)

When the nonlinear function and node numbers of the two connected nodes are known, the vector $R^{Non}$ can be constructed automatically by programming.

2. Matrix calculation for linear flexible connection

After adding linear flexible connections to these subcomponent models, equation (14) can be rewritten as follows

\begin{matrix} [M] {\overset{\cdot\cdot}{u}} & + ([C] + [C^{Lin}]) {\overset{\cdot}{u}} \\ + ([K] + [K^{Lin}]) {u} = {r} + {R^{Non} (\overset{\cdot}{u}, u, t)} \end{matrix}

(18)

Without loss of generality, take such a system as an example: four nodes in total, node 1 and node 3 are coupled through linear flexible connection. The calculation of $K^{Lin}$ is demonstrated. And the same method can be used for the calculation of $C^{Lin}$ .

Define

N^{Lin} = [\begin{matrix} I & 0 & - I & 0 \\ 0 & 0 & 0 & 0 \\ - I & 0 & I & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]

(19)

Therefore

K^{Lin} = [\begin{matrix} k & 0 & 0 & 0 \\ 0 & k & 0 & 0 \\ 0 & 0 & k & 0 \\ 0 & 0 & 0 & k \end{matrix}] * N^{Lin}

(20)

where $N^{Lin}$ is a block matrix with identity matrices at the entries related to the connected nodes, node 1 and node 3; $k$ is the stiffness matrix of linear flexible connection. The bearing and coupling are assumed to possess constant stiffness and damping and can be treated as the linear flexible connections. Their stiffness matrices are expressed as $k = diag (k_{x}, k_{y}, k_{z}, k_{θ x}, k_{θ y}, 0)$ and $k = diag (0, 0, 0, 0, 0, k_{θ z})$ , respectively.

If the number of linear flexible connections is greater than one, add the connections one by one

K^{Lin} = K^{Lin_1} + K^{Lin_2} + \dots + K^{Lin_N}

(21)

When the stiffness matrix $k$ of linear flexible connection and node numbers of the two connected nodes are known, the matrix $K^{Lin}$ can be constructed automatically by programming.

3. Matrix calculation for rigid connection

After adding rigid connections to these subcomponent models, equation (18) can be rewritten as follows

\begin{matrix} [P] [M] {[P]}^{T} {\overset{\cdot\cdot}{U}} + [P] ([C] + [C^{Lin}]) {[P]}^{T} {\overset{\cdot}{U}} \\ + [P] ([K] + [K^{Lin}]) {[P]}^{T} {U} \\ = [P] ({r} + {R^{Non} ({[P]}^{T} \overset{\cdot}{U}, {[P]}^{T} U, t)}) \end{matrix}

(22)

Since the number of effective nodes in the coupling system will be reduced by rigid connections, the generalized displacement in equation (22) is marked by $U$ , which comes from $u$ in equation (18) by deleting some duplicate nodes. Without loss of generality, take such a system as an example: four nodes in total, node 1 and node 3 are coupled through rigid connection. The derivation of matrix calculation for a rigid connection will be demonstrated as follows.

Define

P = P_{2} P_{1} = [\begin{matrix} I & 0 & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & 0 & 0 & I \end{matrix}] [\begin{matrix} I & 0 & I & 0 \\ 0 & I & 0 & 0 \\ 0 & 0 & I & 0 \\ 0 & 0 & 0 & I \end{matrix}]

(23)

Therefore

\begin{matrix} [P_{1}] [K] {[P_{1}]}^{T} = \\ [\begin{matrix} K_{11} + K_{31} + K_{13} + K_{33} & K_{12} + K_{32} & K_{13} + K_{33} & K_{14} + K_{34} \\ K_{21} + K_{23} & K_{22} & K_{23} & K_{24} \\ K_{31} + K_{33} & K_{32} & K_{33} & K_{34} \\ K_{41} + K_{43} & K_{42} & K_{43} & K_{44} \end{matrix}] \end{matrix}

(24)

\begin{matrix} [P_{2}] [P_{1}] [K] {[P_{1}]}^{T} {[P_{2}]}^{T} = \\ [\begin{matrix} K_{11} + K_{31} + K_{13} + K_{33} & K_{12} + K_{32} & K_{14} + K_{34} \\ K_{21} + K_{23} & K_{22} & K_{24} \\ K_{41} + K_{43} & K_{42} & K_{44} \end{matrix}] \\ = [P] [K] {[P]}^{T} \end{matrix}

(25)

{[P]}^{T} {U} = {[P]}^{T} {\begin{matrix} u_{1 p 3} \\ u_{2} \\ u_{4} \end{matrix}} = {\begin{matrix} u_{1 p 3} \\ u_{2} \\ u_{1 p 3} \\ u_{4} \end{matrix}} = {\begin{matrix} u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{matrix}} = u

(26)

where $P_{1}$ comes from adding a 6-by-6 identity matrix at row 1, column 3 of an identity diagonal block matrix. Note that 1 and 3 are the connected node numbers, and the row number should be smaller than the column number. $[P_{1}] [K] [P_{1}]^{T}$ adds row 3 to row 1 and then adds column 3 to column 1. Therefore, node 1 obtains the stiffness of node 3. $P_{2}$ comes from eliminating row 3 of an identity diagonal block matrix. $[P_{2}] [P_{1}] [K] [P_{1}]^{T} [P_{2}]^{T}$ deletes row 3 and column 3 of $[P_{1}] [K] [P_{1}]^{T}$ . Since $R^{Non}$ is the nonlinear function of $u$ , while the generalized displacement of equation (22) is $U$ , the coordinate transformation is shown in equation (26).

If the number of rigid connections is greater than one, add all the connections at a time. Take such a system as an example: five nodes in total, node 1 and node 3 as well as node 2 and node 5 are coupled through rigid connections. The matrix $P$ is demonstrated as follows

P = P_{2} P_{1} = [\begin{matrix} I & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 \\ 0 & 0 & 0 & I & 0 \end{matrix}] [\begin{matrix} I & 0 & I & 0 & 0 \\ 0 & I & 0 & 0 & I \\ 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & I \end{matrix}]

(27)

When the node numbers of all the rigid-connected nodes are known, the matrix $P$ can be constructed automatically by programming. Among the three kinds of connections, the rigid connection will reduce the effective nodes in the coupling system and change the node number, so it is recommended to add rigid connection finally. The order of adding linear and nonlinear connections is interchangeable, as long as it is before adding rigid connection.

Figure 5.

Coupling model of the drum driving system.

Simulation and analyses

Drum load

As illustrated in Figure 6, the dynamic coupling model of the drum driving system is constructed after fixing the $xyz$ coordinates on the ranging arm, and $XYZ$ represents the ground coordinates. The X-axis lies on the ground and is parallel to the shearer hauling direction, and the Y-axis points up in the vertical direction. The drum load includes the three-directional cutting moment ( $T_{X}^{d}$ , $T_{Y}^{d}$ , and $T_{Z}^{d}$ ) and the three-directional cutting force ( $F_{X}^{d}$ , $F_{Y}^{d}$ , and $F_{Z}^{d}$ ). The mean value of drum load is calculated based on the following empirical formula²²

{\begin{matrix} T_{Z}^{d} = T_{N} i η \\ F_{Y}^{d} = T_{Z}^{d} / D \\ F_{X}^{d} = κ_{x} F_{Y}^{d} \\ F_{Z}^{d} = κ_{z} F_{Y}^{d} \end{matrix}

(28)

where $T_{N}$ is the nominal torque of the asynchronous motor, which is equal to 7 742.7 N m; $i$ is the transmission ratio of the transmission system of the cutter, which is equal to 88.1; $η$ is the drum driving system efficiency, which is assumed to be equal to one for a conservative result; $D$ is the drum diameter, which is equal to 4.35 m; $κ_{x}$ and $κ_{z}$ are the parameters related to the structure of the drum; $κ_{x}$ is the hauling resistance coefficient, which has a range of 0.6–0.8; $κ_{z}$ is the axial force coefficient, which has a range of 0–0.2;²² and $T_{X}^{d}$ and $T_{Y}^{d}$ are approximately zero and can be neglected.²³

Figure 6.

Coordinate system of drum load: (a) side view and (b) top view.

Because of the cyclic variation of the number of teeth actually involved in cutting and the nonuniform coal/rock strength, the actual cutting load fluctuates about its mean value. The load fluctuation coefficient, which is defined as the ratio of the peak load to mean load, is set to 1.2 based on the experimental data.²² The time series of the drum load components are presented in Figure 7.

Figure 7.

Time series of drum load: (a) cutting moment component and (b) cutting force component.

Verification of the proposed gear model

In this article, an improved 3D parallel-axis gear model, which has time-varying pressure angle and contact ratio, was proposed to replace Velex’s gear model²¹ used in the authors’ previous work.¹⁷ First of all, we should determine (1) whether the outcomes from the proposed model are reasonable and (2) how much is the difference between outcomes from the proposed model and Velex’s gear model, and also whether these differences are explicable.

Figure 8(a) illustrates the dynamic meshing force of the gear pair Z₂₁–Z₂₂ from the proposed model and Velex’s model. It is obvious that the two curves coincide highly with each other. Figure 8(b) illustrates the dynamic factors of various gear pairs from the two models. The biggest difference appears at the gear pair Z₁₂–Z₁₃, an increase of only 0.87%. This is because the variation of gear center distance is insignificant in our case, and the reasons will be further discussed later.

Figure 8.

Comparison of dynamic meshing force from the proposed model and Velex’s gear model: (a) dynamic meshing force of the gear pair Z₂₁–Z₂₂ and (b) dynamic factor of various gear pairs.

Figure 9(a) illustrates the equivalent mesh misalignment of the gear pair Z₂₁–Z₂₂ from the proposed model and Velex’s model. The two curves are similar in shape, but different in amplitude. This is because the equivalent mesh misalignment can be enhanced due to the differences of contact ratio and meshing phase between different mesh elements along the tooth width. The proposed model takes this effect into consideration, while Velex’s gear model does not. Figure 9(b) illustrates the peak mesh misalignment of various gear pairs from the two models. It is obvious that the increase of peak mesh misalignment for the second-stage gears is larger than that in the first stage (gear pair Z₁₁–Z₁₂ and gear pair Z₁₂–Z₁₃). Particularly, the peak mesh misalignment of the gear pair Z₂₃–Z₂₄ increases from 17.7 to 26.6 μm (an increase by 50.3%). This is because the tooth width of the first stage is 90 mm, while it is 140 mm for the second stage.

Figure 9.

Comparison of equivalent mesh misalignment from the proposed model and Velex’s gear model: (a) equivalent mesh misalignment of the gear pair Z₂₁–Z₂₂ and (b) peak equivalent mesh misalignment of various gear pairs.

The above results illustrate that the proposed 3D parallel-axis gear model is credible, and it is should be used when equivalent mesh misalignment is concerned, especially for wide-faced gear.

Deformation of ranging arm housing

As illustrated in Figure 10, a comparative analysis was conducted between two loading scenarios, that is, M, where only the cutting moment was applied, and M + F, where both the cutting moment and the cutting force were applied. Figure 11(a) illustrates the y-direction deformation of node 1 on the ranging arm housing (the node numbers are presented in Figure 5) under the two loading scenarios. It can be observed that the dynamic displacement of node 1 significantly increases after the three-directional cutting force is applied on the drum. Figure 11(b) presents the root mean square (RMS) values of the deformations of various bearing bores of the ranging arm housing from the motor end to the drum end. In particular, this value increases 386.5 times after the cutting force is applied, implying that the three-directional cutting force is the primary external factor leading to the large deformation of the ranging arm housing. It is also observed that the ranging arm housing deformation displays an increasing trend from the motor end to the drum end, which is indicative of the features of a cantilever.

Figure 10.

Two loading scenarios: (a) M loading scenario and (b) M + F loading scenario.

Figure 11.

Comparison of ranging arm housing deformation under two loading scenarios: (a) Y-direction deformation of bearing bore BP1 on ranging arm housing and (b) RMS deformation of various bearing bores on ranging arm housing.

Effects of housing deformation on the transmission system

The gears, bearings, and elastic shafts are the components of the drum driving system that are likely to malfunction. This section describes a systematic analysis of the effects of a large housing deformation on the dynamic characteristics of these components.

Effects of large deformation of ranging arm housing on gears

As mentioned previously, the parallel-axis gear model in this study considers the influences of the gear center distance variations on the pressure angle and contact ratio. Figure 12 illustrates the center trajectories of the gear pair Z₂₁–Z₂₂ under two loading scenarios, that is, M indicates that only the cutting moment is applied, while M + F represents both the cutting moment and the cutting force are applied. After applying the 3D cutting force, the gear centers moved to the negative direction of the x-axis, the positive direction of the y-axis (Figure 12(a)), and the negative direction of the z-axis (Figure 12(b)), among which the displacement in the x-direction is the smallest. It can also be noticed that gear $Z_{22}$ has a larger translational motion than gear $Z_{21}$ (the numbering of various gear pairs is illustrated in Figure 2). Hence, the obtained results are consistent with the intuition that the cantilever beam has a larger displacement at the free end.

Figure 12.

Trajectories of the gear pair Z₂₁–Z₂₂ under two loading scenarios: (a) trajectories in the x–y plane and (b) trajectories in the z–x plane.

Figure 13(a) illustrates the dynamic center distance of the gear pair Z₂₁–Z₂₂ under the two loading scenarios. The amplitudes of the curves $(Δ)$ are 0.013 and 0.033 mm under the M and M + F loading scenarios, respectively. And the maximum deviations from the nominal center distance $(Γ)$ are 0.018 and 0.020 mm under the M and M + F loading scenarios, respectively. Figure 13(b) illustrates the $Δ$ and $Γ$ values of different gear pairs under the two loading scenarios. It should be noticed that the maximum variation of center distance under two loading scenarios is not more than 0.04 mm, which is negligible when comparing with their nominal values (hundreds of millimeters). This is because the center distance is sensitive to the x-direction (center-line direction) deformation, while it is insensitive to the y- or z-direction deformation. As mentioned previously, the displacement in the x-direction is the smallest.

Figure 13.

Dynamic center distance of gear pairs under two loading scenarios: (a) dynamic center distance of the gear pair Z₂₁–Z₂₂ and (b) comparison of center distance between various gear pairs.

Figure 14(a) illustrates the dynamic contact ratio of the gear pair Z₂₁–Z₂₂ (in the middle of tooth width) under two loading scenarios, the maximum deviation of which is $3.2 \times 10^{- 3}$ (1.6279 – 1.6247). Comparing with the nominal contact ratio, it can be concluded that the large deformation of ranging arm housing does not have a significant influence on the contact ratios of parallel-axis gears. The same conclusion can be obtained for the effect of housing deformation on the dynamic pressure angles. In addition, it should be noted that the gear transmission system is working under the medium speed and heavy load. Therefore, no significant difference can be found between the dynamic meshing force of the gear pair Z₂₁–Z₂₂ under the M and M + F loading scenarios, as shown in Figure 14(b).

Figure 14.

(a) Dynamic contact ratio and (b) dynamic meshing force of the gear pair Z₂₁–Z₂₂ under two loading scenarios.

It is valuable to investigate load distribution over the gear face width. Mesh misalignment is closely related to the load distribution along the face width of gears. Figure 15(a) illustrates the mesh misalignment of the gear pair Z₂₁–Z₂₂ under the M and M + F loading scenarios. The peak mesh misalignment, which is 18.7 μm under the M loading scenario, increases to 26.6 μm under the M + F loading scenario (an increase of 42%). Figure 15(b) illustrates the peak mesh misalignments of various gear pairs under the two loading scenarios. It is observed that the large deformation of the ranging arm housing results in a substantial increase in the mesh misalignment of the transmission system. Figure 16 illustrates the spectra of mesh misalignment of the gear pair Z₂₁–Z₂₂ under the two loading scenarios, wherein a significant discrepancy occurs only at 17 Hz. From Figure 11(a), it can be observed that 17 Hz corresponds to the large deformation of the ranging arm housing. This implies that the increase in the mesh misalignment of the gear pairs is caused by the large deformation of the ranging arm housing. Figure 17 illustrates the load distributions of the gear pair Z₂₁–Z₂₂ along the face width under the M and M + F loading scenarios, respectively, over a duration of two meshing periods. It is obvious that the large deformation of the ranging arm housing leads to more serious uneven load, and a larger maximum load density is found. Therefore, it is necessary to adopt a larger face load factor or conduct a lead modification when designing the gears of the transmission system.

Figure 15.

Mesh misalignment of gear pairs under two loading scenarios: (a) mesh misalignment of the gear pair Z₂₁–Z₂₂ and (b) comparison of peak mesh misalignment between various gear pairs.

Figure 16.

Mesh misalignment spectra of the gear pair Z₂₁–Z₂₂ under two loading scenarios: (a) M loading scenario and (b) M + F loading scenario.

Figure 17.

Load distribution of the gear pair Z₂₁–Z₂₂ under two loading scenarios: (a) M loading scenario and (b) M + F loading scenario.

Effects of large deformation of ranging arm housing on bearings

Bearing is a key component of the transmission system of the cutter. The failure of a bearing can result in the failure of the entire transmission system. Figure 18 illustrates the dynamic load of bearing $B_{6}$ (the numbering of various bearings is illustrated in Figure 2) under the two loading scenarios, and it can be concluded that the dynamic bearing force is not significantly affected by the large deformation of the ranging arm housing. This can be attributed to the negligible difference in the dynamic meshing forces of the gear pair Z₂₁–Z₂₂ between the two loading scenarios, as illustrated in Figure 14(b). Moreover, the frequency of large deformation of the ranging arm housing (17 Hz) is relatively low, generating a relatively small inertia force. However, it can be observed from Figure 19 that a large deformation of the ranging arm housing causes an increase in the coaxial angle of the inner and outer races of the bearing, from $3.15 \times 10^{- 4}$ to $7.69 \times 10^{- 4} rad$ (an increase of 144%), which will promote the unbalanced loading of the bearing roller and adversely affects the longevity of the bearing. Therefore, sufficient attention should be paid to selecting bearings such that the allowed coaxial degree of their inner and outer races satisfies the design requirement.

Figure 18.

Dynamic loads of bearing B₆ under two loading scenarios: (a) M loading scenario and (b) M + F loading scenario.

Figure 19.

Coaxial degree values of inner and outer races of bearing B₆ under two loading scenarios: (a) bearing coordinate and (b) comparison of coaxial angle.

Effects of large deformation of ranging arm housing on elastic shaft

In order to reduce the impact on the transmission system and ensure overload protection, a thin and long elastic shaft is set inside the hollow shaft of the cutting motor, as illustrated in Figure 2. Designing a significantly large safety margin for this elastic shaft will prevent it from breaking when the drum experiences an excessive impact, rendering it incapable of protecting the transmission system. On the other hand, a significantly low design strength for this elastic shaft will result in frequent breaks, affecting the production efficiency. In addition, in order to meet the important trend of unmanned operation in the future mining systems, the torsional vibration of the elastic shaft is used as the feedback signal to reflect the drum load.⁹ However, since the housing deformation is neglected, the accuracy of this approach still needs to be validated. The comparative analysis results illustrated in Figure 20 indicate that a large deformation of the ranging arm housing has no significant impact on the torsional deformation of the elastic shaft. This is because the elastic shaft mainly transfers the torque load. As the dynamic meshing forces are almost the same under the two loading scenarios (see Figure 14(b)), the torque load on the elastic shaft has no obvious change. Therefore, it is not necessary to take the housing deformation into consideration in the design of the elastic shaft.

Figure 20.

Elastic torsional deformation under two loading scenarios: (a) M loading scenario and (b) M + F loading scenario.

Conclusion

In order to accurately calculate the dynamic meshing force of gears suffering housing’s large deformation, a new 3D parallel-axis gear model was constructed considering the time-varying pressure angle and contact ratio. Then, a standard coupling method was adopted to construct the coupling dynamic model for the complete drum driving system, which simultaneously considers the flexibility of the ranging arm housing, bending and torsional deflection of the gear shaft, nonlinear meshing force of the gear pairs, bearing stiffness, load characteristics of the drum, and mechanical properties of the motor. Simulations under two loading scenarios, that is, M (applying only the cutting moment) and M + F (applying both cutting moment and cutting force), allowed the following conclusions to be reached:

Compared with Velex’s gear model,²¹ the equivalent mesh misalignment is larger for the proposed 3D parallel-axis gear model with time-varying pressure angle and contact ratio. The dynamic meshing forces from the two gear models are almost the same, since gear center distance variations are insignificant in our case. Thus, the proposed model is better to be applied when equivalent mesh misalignment is concerned, especially for wide-faced gear.

The three-directional cutting force is the major external factor contributing to a large deformation of the ranging arm housing, which displays an increasing trend from the motor end to the drum end, demonstrating the features of a cantilever. This deformation of the ranging arm housing results in unbalanced loading on both the gears and bearings, which accelerates their damage process; however, it has a negligible impact on the dynamic meshing forces of gears, dynamic loads of bearings, and torsional deformation of the elastic shaft.

Considering the effects of a large deformation of the ranging arm housing on the key components of the transmission system obtained in this study, three guidelines can be recommended for the design of a drum driving system. First, it is necessary to adopt a larger face load factor or conduct a lead modification of gears. Second, it is necessary to select bearings such that the allowed coaxial degree of their inner and outer races satisfies the design requirement. Third, it is not necessary to take housing deformation into consideration in the design of an elastic shaft.

Footnotes

Acknowledgements

The authors would like to acknowledge the support and contribution from the State Key Laboratory of Mechanical Transmissions, Chongqing University, China.

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the National Major Basic Research Program of China (973 Program, Grant No. 2014CB046304).

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