1. Introduction

2. Design of the Simulated Shearer Drum

2.2. Design of the Simulated Drum

As the core cutting component in the coal mining machine's cutting process, designing a similar simulation around the cutting drum is key to ensuring the experimental system and the prototype machine's working conditions are similar. The simulation cutting drum builds processes based on similarity theory is shown in Figure 3. Therefore, this thesis focuses on the drum structure of the coal mining machine as the main research subject, derives similarity criteria through MLT dimensional analysis, and studies the cutting mechanism of the drum during operation, as well as related motion and structural parameters.

Parameters defining the drum's geometric structure as Table 1 include its overall diameter (D), the external diameter of the blade (D_y), the depth of cut made by the drum (B), the angle of elevation for the spiral blade (α_y), the leading distance of the blade (L), the total number of blade heads (Z), the spacing between blades (S_y), the angle of wrap for the blade around the hub (β_y), the spacing of the picks (T_c), the angle at which picks are mounted (γ), and the angle of pick inclination (λ_s). For material parameters, the focus lies on simulating the compressive strength (σ) and the density (ρ) of coal and rock, in line with the criteria for coal and rock identification. Operational parameters encompass the range of the swing angle (θ) for the rocker arm, the rotational velocity of the drum (n), and the speed of traction (v) [14].

Using the MLT basic dimensional system (i.e., mass, length, time), the values and dimensions of the relevant parameters of the prototype are listed. In the design of similar models, dimensionless physical quantities have the same values between the prototype and the model, making it unnecessary to derive related similarity criteria. From the analysis above, it is necessary to derive similarity criteria for a total of 5 parameters as Table 2: D, n, ρ, v, σ. According to the second theorem of similarity (the π theorem), with 5 similarity parameters and 3 basic dimensions, the number of π rules is 2, calculated as 5-3. Dimensional analysis shows that D, n, ρ include the three basic dimensions of M, L, T, and the determinant formed by them is not zero. They correspond respectively to parameters related to the structure of the cutting part, the motion parameters of the coal mining machine, and the characteristics of the cutting material. Therefore, these are selected as the basic physical quantities to list in the dimensional matrix exponent table.

Table 1. Parameters of the simulated shearer cutting system.

Table 1. Parameters of the simulated shearer cutting system.

Parameters	Symbols	Units	M	L	T
Drum diameter	D	mm	0	1	0
Blade outer edge diameter	Dy	mm	0	1	0
Drum cut depth	B	mm	0	1	0
Spiral blade lift angle	αy	°	0	0	0
lead of the blade	L	mm	0	1	0
Blade head number	Z	null	0	0	0
Blade pitch	Sy	mm	0	1	0
Blade hub wrap angle	βy	°	0	0	0
Pick pitch	Tc	mm	0	1	0
Pick installation angle	γ	°	0	0	0
Pick inclination angle	λs	°	0	0	0
Rotating speed of drum	n	r/min	0	0	-1
Traction speed	v	m/min	0	1	-1
Density	ρ	kg/m³	1	-3	0
Compressive strength	σ	Mpa	1	-1	-2

Table 2. Dimensional matrix of spiral drum design parameters.

Table 2. Dimensional matrix of spiral drum design parameters.

Parameters	n	ρ	D	v	σ
Index	a1	a2	a3	a4	a5
M	0	1	0	0	1
L	0	-3	1	1	-1
T	-1	0	0	-1	-2

Use the exponential method to analyze the dimensions of the system and obtain the linear homogeneous equations of the quality system.

$$\{\begin{array}{l}{\mathrm{M}\text{:}\mathrm{a}}_2+{\mathrm{a}}_5=0\hfill \\ \mathrm{L}\text{:}-3{\mathrm{a}}_2+{\mathrm{a}}_3+{\mathrm{a}}_4-{\mathrm{a}}_5=0\hfill \\ \mathrm{T}\text{:}-{\mathrm{a}}_1-{\mathrm{a}}_4-2{\mathrm{a}}_5=0\hfill \end{array}$$

(5)

The similarity criterion for calculating different dimension parameters is:

$$\{\begin{array}{l}{\pi }_1=\frac{\mathrm{v}}{\mathrm{nD}}\hfill \\ {\pi }_2=\frac{\sigma }{{\mathrm{n}}^2\mathsf{\rho }{\mathrm{D}}^2}\hfill \end{array}$$

(6)

The π term is an invariant, and the similarity index is one according to the first similarity theorem. The similarity coefficient expression is as follows:

$$\{\begin{array}{l}{\mathrm{C}}_{\mathrm{n}}{\mathrm{C}}_{\mathrm{D}}={\mathrm{C}}_{\mathrm{v}}\hfill \\ {\mathrm{C}}_{\mathsf{\rho }}{\mathrm{C}}_{\mathrm{v}}^2={\mathrm{C}}_{\sigma }\hfill \end{array}$$

(7)

The similarity coefficient for drum diameter, C_D, requires that the model drum diameter be ≤400mm, therefore C_D is set to 1/3 as Table 3. The simulated cutting material is made from coal dust and rock dust cut by the prototype machine, hence C_ρ equals 1.

Based on the structure of the prototype shearer, it is determined that the simulated cutting drum is configured with teeth arranged in a sequential manner. The number of blade heads on the drum is set to 2, corresponding to 3 cutting lines, with 2 teeth configured on each cutting line. Based on this, establish the model of the cutting drum, as shown in Figure 4.

3. The Experimental Platform Device Structure

4. Software System Design in Digital Space

4.2. Model Layer

The model layer mainly analyzes and stores the following models: the geometric model of the cutting experiment platform, the geometric model of the coal rock sample and the mechanism model of the cutting experiment platform.

4.2.1 Geometric Model

The geometric model refers to a mathematical representation method used to describe the shape and structure of objects. Geometric models are three-dimensional (such as solid objects), aiming to mathematically capture and express the geometric features of objects, enabling computers to process, analyze, render, and simulate them. Geometric models are usually represented by data structures consisting of vertices, edges, and faces, which can construct complex geometric models, from simple geometric bodies to highly detailed 3D models.

The geometric models of the cutting experiment platform mainly include: the X, Y-axis feeding, Z-axis lifting model, and the cutting unit model, as shown in Figure 17. The X, Y-axis feeding, Z-axis lifting model describes the motion control of the cutting device in three orthogonal directions, including the adjustment of feeding speed and lifting speed. The cutting unit model refers to the design and functionality of a single cutting drum that completes the coal rock cutting task.

The coal rock sample model includes: the 3D model of the coal rock sample and the coal rock interface model of the coal rock sample. The 3D model of the coal rock sample represents the three-dimensional appearance of the coal rock sample, containing the shape, size, and internal structure of the coal rock. The coal rock interface model studies the properties of the interface between coal and rock, which is crucial for understanding the mechanical behavior and cutting efficiency during the cutting process.

4.2.2 Mechanism Model

The mechanism model is a type of model used to describe and explain the behavior of a phenomenon or system. It is based on an understanding of the system's internal mechanisms, principles, and interactions, revealing the rules and processes of the system's operation at a microscopic level. Mechanism models are usually established on the basis of a professional field, focusing on the components of the system and their interactions. They describe the dynamic characteristics and behavior of the system through mathematical equations, logical relationships, or graphics. Mechanism models emphasize an understanding of the internal mechanisms and processes of the system, offering stronger interpretability. The mechanism models of the cutting experiment platform include: the single tooth force model, the simulated drum force model, the cutting power transmission model, and the cutting and feeding motor model.

The single tooth force model focuses on the mechanical behavior and force conditions of a single cutting tooth during the coal rock cutting process. The simulated drum force model simulates the force generated by the drum during coal rock cutting, including the drum's dynamic parameters and force. The cutting power transmission model analyzes the energy transfer path from the power source to the cutting head, and how the power is transmitted and acts on the coal rock.

This system transmits the rotary power of the cutting motor through a series of pre-cisely configured transmission shafts (Shafts I to IV) to the main spindle. Taking a single-stage parallel-axis system as an example, the translational-rotational model is presented in Figure 18, and its corresponding dynamical equations are formulated as shown in Equation (8). The lumped parameter model for the multistage gearbox of the Cutting Experimental Platform is shown in Figure 19.

Figure 18. Translational-rotational model of a fixed-shaft gear set.

Figure 18. Translational-rotational model of a fixed-shaft gear set.

Figure 19. Lumped parameter model for the multistage gearbox.

Figure 19. Lumped parameter model for the multistage gearbox.

$$\{\begin{array}{l}{\mathrm{m}}_1{\ddot{\mathrm{x}}}_1=-\left({\mathrm{k}}_{12}{\mathsf{\delta }}_{12}{+\mathrm{c}}_{12}{\dot{\mathsf{\delta }}}_{12}\right){\sin \mathsf{\alpha }}_{12}{-\mathrm{k}}_{\mathrm{x}1}{\mathrm{x}}_1{-\mathrm{c}}_{\mathrm{x}1}{\dot{\mathrm{x}}}_1\\ {\mathrm{m}}_1{\ddot{\mathrm{y}}}_1=\left({\mathrm{k}}_{12}{\mathsf{\delta }}_{12}{+\mathrm{c}}_{12}{\dot{\mathsf{\delta }}}_{12}\right){\cos \mathsf{\alpha }}_{12}{-\mathrm{k}}_{\mathrm{y}1}{\mathrm{y}}_1{-\mathrm{c}}_{\mathrm{y}1}{\dot{\mathrm{y}}}_1\\ {\mathrm{m}}_2{\ddot{\mathrm{x}}}_2=\left({\mathrm{k}}_{12}{\mathsf{\delta }}_{12}{+\mathrm{c}}_{12}{\dot{\mathsf{\delta }}}_{12}\right){\sin \mathsf{\alpha }}_{12}{-\mathrm{k}}_{\mathrm{x}2}{\mathrm{x}}_2{-\mathrm{c}}_{\mathrm{x}2}{\dot{\mathrm{x}}}_2\\ {\mathrm{m}}_2{\ddot{\mathrm{y}}}_2=-\left({\mathrm{k}}_{12}{\mathsf{\delta }}_{12}{+\mathrm{c}}_{12}{\dot{\mathsf{\delta }}}_{12}\right){\cos \mathsf{\alpha }}_{12}{-\mathrm{k}}_{\mathrm{y}2}{\mathrm{y}}_2{-\mathrm{c}}_{\mathrm{y}2}{\dot{\mathrm{y}}}_2\\ {\mathrm{J}}_1{\ddot{\mathsf{\theta }}}_1{=\mathrm{T}}_1-\left({\mathrm{k}}_{12}{\mathsf{\delta }}_{12}{+\mathrm{c}}_{12}{\dot{\mathsf{\delta }}}_{12}\right){\mathrm{r}}_1\\ {\mathrm{J}}_2{\ddot{\mathsf{\theta }}}_2=\left({\mathrm{k}}_{12}{\mathsf{\delta }}_{12}{+\mathrm{c}}_{12}{\dot{\mathsf{\delta }}}_{12}\right){\mathrm{r}}_2{-\mathrm{T}}_2\\ {\mathsf{\delta }}_{12}=\left({\mathrm{x}}_1{-\mathrm{x}}_2\right){\sin \mathsf{\alpha }}_{12}-\left({\mathrm{y}}_1{-\mathrm{y}}_2\right){\cos \mathsf{\alpha }}_{12}{+\mathrm{r}}_1{\mathsf{\theta }}_1{-\mathrm{r}}_2{\mathsf{\theta }}_2{-\mathrm{e}}_{12}\left({\mathsf{\theta }}_1{,\mathsf{\theta }}_2\right)\\ {\mathrm{e}}_{12}\left({\mathsf{\theta }}_1{,\mathsf{\theta }}_2\right){=\mathrm{E}}_{12}\sin \left({\mathrm{Z}}_1{\mathsf{\theta }}_1{+\mathsf{\zeta }}_{12}\right){+\mathrm{E}}_1\sin \left({\mathsf{\theta }}_1{+\mathsf{\eta }}_1\right){+\mathrm{E}}_2\sin \left({\mathsf{\theta }}_2{+\mathsf{\eta }}_2{+\mathsf{\alpha }}_{12}\right)\end{array}$$

(8)

The variable r_i denotes the base circle radius of gear i (where i=1, 2); θ_i represents the rotational angle of the gear; k₁₂, C₁₂, e₁₂, and α₁₂ respectively signify the time-varying mesh stiffness, mesh damping, cumulative mesh error, and mesh angle of the gear pair. k_xi, k_yi, c_xi, and c_yi correspond to the radial support stiffness and damping in the x and y directions for gear i; T_i refers to the torque acting on gear i. E₁₂ and E_i respectively represent the amplitudes of the meshing frequency error for the gear pair and the rotational frequency error of gear i; ζ₁₂ and η_i are the initial phases of the meshing frequency error and rotational frequency error, respectively; δ₁₂ is the meshing deformation on the tooth surface engagement line considering the comprehensive error.

The equations of motion for the cutting motor model, the cutting drive system dynamic model, and the drum load model are compiled and organized into matrix form. This yields the electromechanical coupled system dynamics mathematical model for the cutting section, as shown in Equation (9).

$$\mathrm{M} \ddot{\mathrm{X}}+\left({\mathrm{C}}_{\mathrm{m}}{+\mathrm{C}}_{\mathrm{t}}\right)\dot{\mathrm{X}}+\left({\mathrm{K}}_{\mathrm{m}}{+\mathrm{K}}_{\mathrm{t}}{+\mathrm{K}}_{\mathrm{b}}\right){\mathrm{X}=\mathrm{T}}_{\mathrm{L}}{+\mathrm{T}}_{\mathrm{e}}$$

(9)

In the equation, X represents the generalized coordinate vector, with X = [x_i y_i θ_i], where i = m, 1, 2, ..., 5, d; M, T_L, and T_e are the generalized mass matrix, system load vector, and the electromagnetic torque vector of the cutting motor, respectively. K_m, K_t, and K_b respectively represent the mesh stiffness matrix, torsional stiffness matrix, and bearing stiffness matrix, respectively; C_m and C_t represent the mesh damping matrix and torsional damping matrix, respectively. The equations in Equation (9) describe a system of second-order differential equations for the mechanical transmission system. For ease of computation, it is first necessary to reduce the order, transforming it as follows:

$$\{\begin{array}{l}\frac{\mathrm{d}}{\mathrm{dt}}\mathrm{X}=\dot{\mathrm{X}}\\ \frac{\mathrm{d}}{\mathrm{dt}}\dot{\mathrm{X}}=-{\mathrm{M}}^{-1}\left({\mathrm{C}}_{\mathrm{m}}+{\mathrm{C}}_{\mathrm{t}}\right){\dot{\mathrm{X}}-\mathrm{M}}^{-1}\left({\mathrm{K}}_{\mathrm{m}}{+\mathrm{K}}_{\mathrm{t}}{+\mathrm{K}}_{\mathrm{b}}\right)\mathrm{X}\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\mathrm{M}}^{-1}\left({\mathrm{T}}_{\mathrm{L}}+{\mathrm{T}}_{\mathrm{e}}\right)\end{array}$$

(10)

This can further be expressed in matrix form:

$$\begin{array}{l}\left\{\begin{array}{l}\dot{\mathrm{X}}\hfill \\ \ddot{\mathrm{X}}\hfill \end{array}\right\}=\left[\begin{array}{cc}\mathrm{I}& 0\\ -{\mathrm{M}}^{-1}\left({\mathrm{C}}_{\mathrm{m}}+{\mathrm{C}}_{\mathrm{t}}\right)& -{\mathrm{M}}^{-1}\left({\mathrm{K}}_{\mathrm{m}}+{\mathrm{K}}_{\mathrm{t}}+{\mathrm{K}}_{\mathrm{b}}\right)\end{array}\right]\left\{\begin{array}{l}\mathrm{X}\hfill \\ \dot{\mathrm{X}}\hfill \end{array}\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}+\left\{\begin{array}{c}0\\ {\mathrm{M}}^{-1}\left({\mathrm{T}}_{\mathrm{L}}+{\mathrm{T}}_{\mathrm{e}}\right)\end{array}\right\}\end{array}$$

(11)

Based on the aforementioned mathematical model, simulation models of the cutting motor and the cutting drive system are constructed separately on the MATLAB/Simulink platform, as shown in Figure 20. The angular displacement and angular velocity of the cutting motor are used as shared variables to transfer data between the cutting motor and the cutting drive system. This data is then used to calculate in real time the torsional load on the motor output shaft, which is directly fed back to the cutting motor. Consequently, this process establishes an electromechanical coupled simulation model for the cutting drive system.

During computation, the ode45 solver provided in MATLAB is utilized (employing the fourth-fifth order Runge-Kutta method, which uses a fourth-order method to generate candidate solutions and a fifth-order method to control errors, constituting an adaptive step-size numerical solution technique for ordinary differential equations). This allows for the solving of the system's differential equations, thereby obtaining the dynamic response of each component of the system.

4.4. Digital Twin System Interaction Interface

Experimental platform digital twin systems are created using Unity3D, where three-dimensional models are imported and then driven based on sensor data. The steps for importing 3D models are as follows: First, prepare the model file and use 3ds Max software to convert the experimental platform model into an fbx format supported by Unity3D, as shown in Figure 22.

Then, drag the model file into the Assets folder of the Unity3D editor to complete the import of the model file. Unity3D will automatically process the imported model. Next, drag the imported model from the Assets folder to the scene, and adjust the model's position, rotation angle, and scale to meet the requirements of the scene.

Programming is required to process sensor data. First, determine the type and interface of the sensor used. In Unity3D, C# scripts can be written to read sensor data. In addition, the WebSocket library's API is used to read sensor data transmitted over the network. Based on the obtained sensor data, Unity3D's Transform component is used to adjust the position and rotation angle of the experimental platform model. The Digital Twin Experimental Platform's system interaction is shown in Figure 23.

Figure 22. (a) Expanded UV grid; (b) UV mapping; (c) heat map rendering.

Figure 22. (a) Expanded UV grid; (b) UV mapping; (c) heat map rendering.

Figure 23. Digital Twin Experimental Platform System Interaction Interface.

Figure 23. Digital Twin Experimental Platform System Interaction Interface.

Experimental platform digital twin system also integrates visible light and infrared video camera streams. By using Unity's WebCamTexture class, a WebCamTexture object can be created and the camera's name specified. Then, assign the WebCamTexture object to the Material's Texture property to display the video on a GameObject in the scene. In addition, control of the WebCamTexture's playback and pause is required.

The system uses the LineRenderer component to display a dynamic curve graph of real-time sensor data. The operation process is as follows: Create an empty GameObject in the Unity editor and add a LineRenderer component to it. Then, write a new C# script to control the updating and rendering of the curve data. In the script, calculate the position of each point on the curve based on the dynamically updated data, and use the LineRenderer's SetPositions() method to update these points, thereby achieving dynamic updating of the curve.

5. Simulated Cutting Experiments

6. Conclusions

References

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