# Dynamic Response Analysis of the Coal Gangue-like Elastic Rock Sphere Impact on the Massless Tail Beam Based on Contact-Structure Theory and FEM

Authors: Yang Yang; Lirong Wan; Zhengyuan Xin
Journal: Shock and Vibration
 (2019)

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## Abstract

The impact and collision behavior between the coal gangue and the hydraulic support widely exists in the top coal caving. However, due to the complex interactions between the large number of coal gangue particles in the mining surface and the limitations of the coal mining technology and other factors, it becomes a difficult problem to study the collision behavior and the contact response between the coal gangue and the hydraulic support or its main components under the actual caving conditions. In order to accurately grasp the contact response law when the coal gangue impacts the tail beam of the hydraulic support, in combination with the equivalent stiffness of the tail beam jack, the Lankarani–Nikravesh (L-N) nonlinear spring-damping contact model, the structural mechanics model of the tail beam, and the energy conservation law, this paper firstly establishes the system contact-structure dynamic model when the particles impact on the tail beam based on the tail beam equivalent kinematics model. Then, to further study the system contact response, the spring damper module is used in the finite element software for the first time to replace the hydraulic cylinder, and four different types of the rigid-flexible coupling simulations when the impact heights, the impact positions, the rock radii, and the rock materials change are conducted, respectively. Through the combination of the theory and the simulation, the contact response law when the particles impacting the massless tail beam under different working conditions is obtained, and the system contact response differences as well as the coal gangue identifying feasibility on the basis of the response differences after the coal gangue impact are analyzed. The conclusions will provide theoretical reference and simulation method for the study of the impact-contact behavior between the coal gangue and the hydraulic support.


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## Body

## 1. Introduction

China is a country with coal as its primary resource [1–5]. China’s economy has entered a new normal [6] since the 13th Five-Year Plan was implemented and the coal output has slowed down. However, from the coal production of China from 2008 to 2018 in Figure 1 [7–17], China’s raw coal output still reached 3.68 billion tons in 2018, and the total mined production of the coal remains very large. Meanwhile, according to China’s primary energy consumption structure prognostic figure Figure 2 [6, 18], coal proportion by 2030 will continue to account for more than 60 percent, and it will maintain its dominant position that is irreplaceable for a long time. Moreover, as the important technology of coal mining and the key technology of thick coal seam mining, the study of the top coal caving technology [19–22] is of great significance to improve the mining efficiency and realize the digital mine construction.Figure 1
Energy production of China.Figure 2
Energy consumption structure of China in 2030.As the important mining method, the top coal caving technology and its correlation technique have been studied by plenty of scholars. Yu et al. [23, 24] investigated the fracturing and caving of overburden, put forward the statistical formula of the failure zone maximum heights in the longwall top coal caving (LTCC), and studied the overburden fracturing by the microseismic monitoring. Cao et al. [25] studied the microseismicity in the LTCC through the Flac3d numerical modelling. In the mine which is mining through the combination of a fully mechanized traditional longwall and LTCC methods, Basarir et al. [26] researched the stress of the main and tail gates by the three-dimensional finite difference technique. Based on the in situ conditions in the top coal caving, Alehossein and Poulsen [27] provided a yield and caveability criterion. Zhang et al. [28] investigated the roof failure characteristics, forces acting on supports, and action characteristics of abutment stress of a LTCC working face. Zhang et al. [29] studied the flow of top coal and roof rock and analyzed the loss of top coal in fully mechanized top coal caving mining of extra thick coal seams. Zhang et al. [30] investigated on the coal drawing from thick steep seam with LTCC mining. Liu et al. [31] researched the roof strata conditions, shield pressure, and ground surface subsidence at a LTCC Working Face. Khanal et al. [32] using COSFLOW conducted the mine-scale analysis of LTCC. Si et al. [33, 34] measured the seam gas pressure, gas composition, and ventilation environment in the multilevel LTCC of ultrathick coal seams, and in order to model the gas emissions around the LTCC panels, they established a two-way sequential coupling of a geomechanical and a reservoir simulator. These studies have provided the research basis for the analysis of the mining characteristics of the top coal caving and for the development of the top coal caving technology; however, they rarely refer to the study of the impact behavior and the interaction problem between coal gangue particles and the hydraulic support in the process of top coal caving. The contact response caused by the action of coal gangue particles on the hydraulic support has also not been further explored.The collision and contact behavior between the coal gangue particles and the hydraulic support in the top coal caving process will trigger the vibration of the hydraulic support and the disorderly collision in the coal gangue rebound process, which is of great significance to clearly understand the dynamic behavior of the hydraulic support in the application process and the coal gangue interface identification in the top coal caving process. However, the structure of the hydraulic support is complex, and the repeated impact and mutual disturbance of particles exist in the process of coal gangue particles impacting the hydraulic support, which increases the difficulty of studying the collision behavior between coal gangue particles and the hydraulic support. Therefore, as the main coal drawing component of the top coal caving hydraulic support (as shown in Figure3), the tail beam is taken as the research object in this paper to study the contact response when the single coal gangue impacts the tail beam, so as to pave the way for the later study on the impact and collision behavior between coal gangue particles and the hydraulic support.Figure 3
Hydraulic support in longwall top coal caving.During the collision process between coal gangue and the tail beam, the contact behavior between particles and the tail beam will lead to the complex interlocking responses such as the compression deformation of the particle and the tail beam metal plate, the compression of the liquid within the tail beam jack under the force, and the rotation of the tail beam. Brake proposed an analytical elastic-plastic contact model in the normal direction between two round surfaces in 2012 [35], and then, he developed a new contact model for the normal and oblique impacts with the consideration of strain hardening and frictional effects [36]. Minamoto and Kawamura [37] studied the direct central impact of two identical spheres in the higher speed range. To derive force-displacement relations, Olsson and Larsson [38] studied the tangential contact between an elastic-plastic sphere and a rigid plane analytically and numerically. Wang et al. [39] presented a point contact model of a rigid ball and an elastic-coated solid. Rojek et al. [40] investigated the contact model for the normal interaction between elastoplastic spherical discrete elements. Jackson et al. [41] presented a different methodology for modeling the impact between elastoplastic spheres and compared the several different predicted models. Yang et al. [42, 43] studied the vertical elastic impact dynamic response between the spherical rock and the metal plate and the vertical impact-contact response between the elastic sphere and elastic half space. Wang et al. [44] established a new nonlinear hysteretic model with considering the loading, unloading, and reloading processes. Yan et al. [45] investigated the influence of the falling rock’s shape and impact poses on the impact force and response of RC slabs. Skrinjar et al. [46] compared the evolution of the contact force during the contact process for the general contact-force models and the cylindrical contact-force models. Willert et al. [47] studied the influence of the Tabor parameter on the spheres adhesive normal impact. Zhao et al. [48] investigated the contact problem between the power-law hardening elastic-plastic sphere and the rigid flat under the combination of the normal and tangential loads. Wang et al. [49] conducted the research about the interparticle contact behaviors and microcontact models. Lu et al. [50] studied the contact behavior between the rigid flat and the coated asperity. Yuan et al. [51] established the mechanical model of contact between a sphere-based fractal rough surface and a rigid flat surface. Wang et al. [52] analyzed the mechanical response mechanism when the rockfall impacted the soil cushion layer. Indeed, these research studies on the contact problem or the rock impact problem provided the basic theoretical models, the numerical simulation methods, and the references for the study of impact-contact behavior between coal gangue particles and the tail beam. But they have only referred to the contact behavior between spheres and between the sphere and the plane, or the study of the collision between the sphere and the soil mass. Impact-contact systems are simple. The compression of the liquid under the force and the force transfer of multibody composite structure are not involved. Liang et al. [53] and Wan et al. [54] studied the response when the hydraulic-powered support supported the impacting load; however, they just added the self-defined force on the hydraulic support, the impact and contact process between the coal gangue and the hydraulic support was ignored, and the contact theory when the coal gangue impacts the hydraulic support has not been established yet.On the basis of the previous research results and research deficiencies, to clarify the contact response law of the system after the impact of coal gangue particles on the tail beam of the hydraulic support and grasp the response difference of the system when the coal gangue particles impact the tail beam, respectively, based on the L-N contact model, the kinematics model of the tail beam device, the rotational model of the tail beam impacted by coal gangue particles, and the mechanical model of the tail beam impacted by coal gangue particles is constructed. Then, combining with the contact mechanics and structural mechanics, the dynamic model of the coal gangue particles impacting the tail beam is finally established. Besides, the tail beam owns large volume and mass while the selected particles produced slight impact-contact force. In order to accurately obtain the response of the tail beam under the contact force produced by the impact of the coal gangue particles, the gravity of the tail beam is not considered in this paper. On this basis, the spring damper module is used to replace the hydraulic cylinder in the finite element software for the first time, and then, the finite element simulation model of the coal gangue particles elastic impact on the massless tail beam is constructed. Different types of dynamics simulation of finite element contact are carried out, and the effects of the impact height, the impact position, the rock radii, and the material properties on the system contact response are analyzed.The remainder of the paper is organized as follows: Section2 establishes the kinematics model of the tail beam device. Section 3 constructs the rotational model and the mechanical model of the tail beam impacted by coal gangue particles and finally establishes the dynamic model of the coal gangue particles impacting the tail beam. Section 4 introduces the establishing process of the simulation model, the simulation settings, and the simulation types. Section 5 analyzes the influence of the impact height, the impact position, the rock radii, and the material properties on the system contact response. Section 6 shows some related work and our conclusion.


## 2. Kinematics Model of the Tail Beam Device

In this project, the tail beam of hydraulic support is counted as the research object. The supporting effect of the shield beam and other components of the hydraulic support on the tail beam is equivalent to that of the rigid support frame. As a result, as shown in Figure4, a simplified model of the tail beam device as well as a two-dimensional simplified rod diagram of the tail beam can be obtained (the unit of the length in Figure 4 is mm).Figure 4
Diagram of the tail beam.The size of the tail beam remains constant during the movement process. After the simplification of Figure4, the kinematics diagram of the tail beam (see Figure 5) can be obtained. And if A-B-F-G-H is taken as the vector ring, the equation of closed-loop can be expressed as follows:(1)AB⇀+BF⇀=AH⇀+HG⇀+GF⇀,Figure 5
Kinematics diagram of the tail beam.It is assumed that the angle between AB (A is the center point of the hinged joint hole between the tail beam and shield beam; B is the center point of the hinged joint hole between the tail beam and the jack) isθ. The effective length of the liquid column HG is s (variate). The angle ∠ABF is assumed to be α, the angle between the jack and the horizontal direction is β, and the angle between the upper surface of the tail beam CD and the horizontal direction is ξ. It can be obtained as follows:(2)AB⋅cosθ+BF⋅cosπ−α−β−θ=AH+HG+GF⋅cosβ,BF⋅sinπ−α−β−θ−AB⋅sinθ=AH+HG+GF⋅sinβ.


## 3. Dynamic Model of the Coal Gangue Particles Impacting the Tail Beam

### 3.1. Force Analysis of the Model

As shown in Figure6(a), the tail beam and its jack reach its equilibrium state. It is assumed that the equivalent stiffness of the jack of tail beam is KQ after reaching the equilibrium state and the angle between the jack and the horizontal direction is β1. Then, in accordance with the equivalent stiffness calculating formula [55] of the tail beam, it can be expressed as follows:(3)KQ=kq×kgkq+kg=2SεEgS+2LkyεEg,where kq is the stiffness of the hydraulic oil (high pressure emulsion) inside the jack of the tail beam and kg is the stiffness of the jack cylinder. S is the effective bearing area of the emulsion, ε and Eg are the thickness and elastic modulus of the cylinder, respectively. L is the height of liquid column of hydraulic oil inside the jack of the tail beam, and ky is the volume compression coefficient of the hydraulic oil:(4)KQ=2SεEgEpSEp+2LεEg,where Ep is the volume elastic modulus of the hydraulic oil and Ep=1/ky.Figure 6
Process of coal gangue particles impacting the tail beam. (a) Before the impact. (b) Critical end state of shock compression.
(a)
(b)During the impact of coal gangue particles on the tail beam, a great contact force will be produced by the contact behavior. As a result, it will lead to some changes such as the compression deformation of the coal gangue particles, the deformation of the metal plate subjected to the impact, the elastic compression of the hydraulic oil in the jack of the tail beam, and the downward rotation of the tail beam around hinge point C (see Figure6(b)). In order to simplify the analysis, the change of the equivalent stiffness of the jack caused by the gravity of tail beam is ignored, as well as the change of jack stiffness during the impact of the coal gangue particles on the tail beam. In addition, the deformation of the metal plate subjected to the impact, the cross slip of particles along the metal plate in the process of coal gangue particles contacting with the metal plate is also neglected.Based on the nonlinear relationship between contact force P and compression deformationδδ, Hertz established the contact theory of sphere in contact with other objects [56–60]:(5)P=K⋅δδn,K=4R3⋅E,1E=1−μ12E1+1−μ22E2,1R=1R1+1R2,pr=3P2πa21−ra21/2,a2=R⋅δδ,where n is the nonlinear coefficient, n=3/2. E1, μ1, E2, and μ2 are the elastic modulus and the Poisson’s ratio of two contact objects, respectively. E is the equivalent elastic modulus, R is the equivalent contact radius, E is the contact stiffness, and pr is the contact stress of the contact point. a is the radius of the contact zone. r is the distance between the contact center and the contact point. δδ is the compression of the sphere. R1 and R2 are the contact radii of two contact objects, respectively, and R = R1 for the situation that the sphere impacting the plate.Hertz contact theory ignores the damping force caused by the material damping and the effect of the energy losses of damping on the system contact process, which only applies to the ideal elastic contact conditions. On the basis of Hertz theory, Lankarani and Nikravesh [61, 62] improved the definition of the contact force and established the nonlinear spring-damping contact force models which include the Hertz ideal contact force and the dissipative force of damping. The model can be expressed as [63–67](6)P=K⋅δδn+K⋅δδn⋅Dn⋅δδ⋅,(7)Dn=31−e24δδ0⋅,where Dn is the damping coefficient, δδ0⋅ is the initial velocity of elastic sphere, δδ⋅ is the velocity of sphere, and e is the restitution coefficient.The L-N contact model is one of the most popular and straightforward contact force models, which well conforms with experimental results. It is appropriate for analyzing the contact process of coal gangue particles impacting the metal plate of the tail beam. Therefore, this paper chooses the L-N contact model to calculate the contact force.


### 3.2. Rotational Model of the Tail Beam Impacted by Coal Gangue Particles

The tail beam is rotated downward to the limit position after the impact of coal gangue particles. Then, the rotation angle is defined asγ and the length of the effective liquid column inside the jack before and after the rotation of the tail beam is defined as HG1 and HG2, respectively. AF is changed from AF1 to AF2 in the rotation process, and the angle between the jack and the horizontal direction is changed to β2.(8)AB⋅cosθ+BF⋅cosπ−α−γ−β2+θ=AH+HG2+GF⋅cosβ2,BF⋅sinπ−α−γ−β2+θ−AB⋅sinθ=AH+HG2+GF⋅sinβ2.As is shown in Figure7, the triangle ABF, whose side length AF changes, is taken as the research object. It can be obtained according to Figure 6 and the variation conditions that(9)AF1=AB−BF⋅cosα2+BF⋅sinα2,AF2=AB−BF⋅cosα−γ2+BF⋅sinα−γ2,Δs=AF1−AF2=HG1−HG2,β2=arctanBF⋅sinα−γAB−BF⋅cosα−γ−θ.Figure 7
Movement triangle.


### 3.3. Mechanical Model of the Tail Beam Impacted by Coal Gangue Particles

The tail beam is rotated downward to the limit position after the impact of coal gangue particles, which achieve a force balance under the actions of shield beam and the jack of tail beam (as is shown in Figure8). The equilibrium equation of tail beam dynamics is obtained:(10)FABx=FCon⋅sinπ2−ξ−γ+mg+FAF⋅sinβ2,FCon⋅cosπ2−ξ−γ−FABy=FAF⋅cosβ2,∑MBF=FAF⋅cosπ2−θ−β2⋅BF′+FAF⋅sinπ2−θ−β2⋅BB′,∑MBQ=−FCon⋅sinπ2−ψ+mg⋅sinπ2−ξ−γ+ψ⋅Γ−mg⋅R−δ⋅cosξ+γ,∑MBF+∑MBQ=0,BF′=AB−BF⋅cosα−γ,BB′=BF⋅sinφ,FAF=2KQ⋅Δs,where δ is the maximum compression of the particles when coal gangue particles show vertical impact on the metal plate and lead the tail beam rotates downward to the limit position. FCon is the maximum contact force between coal gangue particles and the metal plate.Figure 8
Mechanical model of the tail beam impacted by coal gangue particles (limit position).


### 3.4. Dynamic Model of the Coal Gangue Particles Impacting the Tail Beam

As shown in Figure9, according to equations (6) and (7), the contact force generated in the contact and compression processes between the rock sphere and the tail beam during the impact is as follows:(11)FCon=K⋅δ3/2⋅1+Dn⋅δ˙.Figure 9
Dynamic model of the system.The jack of the tail beam is replaced by an equivalent spring with the same stiffness, so the changes of mass center and gravitational potential energy during the compression process of the jack can be ignored. It is defined that the height between the initial position where coal gangue particles fall freely from and the position where coal gangue particles initially contact with the upper surface of the tail beam and ish, and the velocity of the particles in the initial contact is as follows:(12)v=2gh,(13)v⊥=2gh⋅sinξ.When the rock sphere impacts the tail beam and is compressed, the change of its potential energy is obtained:(14)ECS=mg⋅R⋅cosξ−Γ⋅cosξ−ψ−R⋅cosξ+γ−Γ⋅cosξ+γ−ψ+δ⋅cosξ+γ.According to the law of energy conservation of the system,(15)
1
2
m
⋅
v
⊥
2
+
E
C
S
=
∫
0
δ
F
Con
d
δ
+
2
⋅
1
2
K
Q
⋅
Δ
s
2
.On the basis of equations (4), (8), (10), (11), (13), (14),and (15), the system contact response when the rock sphere impacting the tail beam is closely related to various parameters, such as the mass and properties of the impact object, the impact height, the contact position, and the initial position of the tail beam.


## 3.1. Force Analysis of the Model

As shown in Figure6(a), the tail beam and its jack reach its equilibrium state. It is assumed that the equivalent stiffness of the jack of tail beam is KQ after reaching the equilibrium state and the angle between the jack and the horizontal direction is β1. Then, in accordance with the equivalent stiffness calculating formula [55] of the tail beam, it can be expressed as follows:(3)KQ=kq×kgkq+kg=2SεEgS+2LkyεEg,where kq is the stiffness of the hydraulic oil (high pressure emulsion) inside the jack of the tail beam and kg is the stiffness of the jack cylinder. S is the effective bearing area of the emulsion, ε and Eg are the thickness and elastic modulus of the cylinder, respectively. L is the height of liquid column of hydraulic oil inside the jack of the tail beam, and ky is the volume compression coefficient of the hydraulic oil:(4)KQ=2SεEgEpSEp+2LεEg,where Ep is the volume elastic modulus of the hydraulic oil and Ep=1/ky.Figure 6
Process of coal gangue particles impacting the tail beam. (a) Before the impact. (b) Critical end state of shock compression.
(a)
(b)During the impact of coal gangue particles on the tail beam, a great contact force will be produced by the contact behavior. As a result, it will lead to some changes such as the compression deformation of the coal gangue particles, the deformation of the metal plate subjected to the impact, the elastic compression of the hydraulic oil in the jack of the tail beam, and the downward rotation of the tail beam around hinge point C (see Figure6(b)). In order to simplify the analysis, the change of the equivalent stiffness of the jack caused by the gravity of tail beam is ignored, as well as the change of jack stiffness during the impact of the coal gangue particles on the tail beam. In addition, the deformation of the metal plate subjected to the impact, the cross slip of particles along the metal plate in the process of coal gangue particles contacting with the metal plate is also neglected.Based on the nonlinear relationship between contact force P and compression deformationδδ, Hertz established the contact theory of sphere in contact with other objects [56–60]:(5)P=K⋅δδn,K=4R3⋅E,1E=1−μ12E1+1−μ22E2,1R=1R1+1R2,pr=3P2πa21−ra21/2,a2=R⋅δδ,where n is the nonlinear coefficient, n=3/2. E1, μ1, E2, and μ2 are the elastic modulus and the Poisson’s ratio of two contact objects, respectively. E is the equivalent elastic modulus, R is the equivalent contact radius, E is the contact stiffness, and pr is the contact stress of the contact point. a is the radius of the contact zone. r is the distance between the contact center and the contact point. δδ is the compression of the sphere. R1 and R2 are the contact radii of two contact objects, respectively, and R = R1 for the situation that the sphere impacting the plate.Hertz contact theory ignores the damping force caused by the material damping and the effect of the energy losses of damping on the system contact process, which only applies to the ideal elastic contact conditions. On the basis of Hertz theory, Lankarani and Nikravesh [61, 62] improved the definition of the contact force and established the nonlinear spring-damping contact force models which include the Hertz ideal contact force and the dissipative force of damping. The model can be expressed as [63–67](6)P=K⋅δδn+K⋅δδn⋅Dn⋅δδ⋅,(7)Dn=31−e24δδ0⋅,where Dn is the damping coefficient, δδ0⋅ is the initial velocity of elastic sphere, δδ⋅ is the velocity of sphere, and e is the restitution coefficient.The L-N contact model is one of the most popular and straightforward contact force models, which well conforms with experimental results. It is appropriate for analyzing the contact process of coal gangue particles impacting the metal plate of the tail beam. Therefore, this paper chooses the L-N contact model to calculate the contact force.


## 3.2. Rotational Model of the Tail Beam Impacted by Coal Gangue Particles

The tail beam is rotated downward to the limit position after the impact of coal gangue particles. Then, the rotation angle is defined asγ and the length of the effective liquid column inside the jack before and after the rotation of the tail beam is defined as HG1 and HG2, respectively. AF is changed from AF1 to AF2 in the rotation process, and the angle between the jack and the horizontal direction is changed to β2.(8)AB⋅cosθ+BF⋅cosπ−α−γ−β2+θ=AH+HG2+GF⋅cosβ2,BF⋅sinπ−α−γ−β2+θ−AB⋅sinθ=AH+HG2+GF⋅sinβ2.As is shown in Figure7, the triangle ABF, whose side length AF changes, is taken as the research object. It can be obtained according to Figure 6 and the variation conditions that(9)AF1=AB−BF⋅cosα2+BF⋅sinα2,AF2=AB−BF⋅cosα−γ2+BF⋅sinα−γ2,Δs=AF1−AF2=HG1−HG2,β2=arctanBF⋅sinα−γAB−BF⋅cosα−γ−θ.Figure 7
Movement triangle.


## 3.3. Mechanical Model of the Tail Beam Impacted by Coal Gangue Particles

The tail beam is rotated downward to the limit position after the impact of coal gangue particles, which achieve a force balance under the actions of shield beam and the jack of tail beam (as is shown in Figure8). The equilibrium equation of tail beam dynamics is obtained:(10)FABx=FCon⋅sinπ2−ξ−γ+mg+FAF⋅sinβ2,FCon⋅cosπ2−ξ−γ−FABy=FAF⋅cosβ2,∑MBF=FAF⋅cosπ2−θ−β2⋅BF′+FAF⋅sinπ2−θ−β2⋅BB′,∑MBQ=−FCon⋅sinπ2−ψ+mg⋅sinπ2−ξ−γ+ψ⋅Γ−mg⋅R−δ⋅cosξ+γ,∑MBF+∑MBQ=0,BF′=AB−BF⋅cosα−γ,BB′=BF⋅sinφ,FAF=2KQ⋅Δs,where δ is the maximum compression of the particles when coal gangue particles show vertical impact on the metal plate and lead the tail beam rotates downward to the limit position. FCon is the maximum contact force between coal gangue particles and the metal plate.Figure 8
Mechanical model of the tail beam impacted by coal gangue particles (limit position).


## 3.4. Dynamic Model of the Coal Gangue Particles Impacting the Tail Beam

As shown in Figure9, according to equations (6) and (7), the contact force generated in the contact and compression processes between the rock sphere and the tail beam during the impact is as follows:(11)FCon=K⋅δ3/2⋅1+Dn⋅δ˙.Figure 9
Dynamic model of the system.The jack of the tail beam is replaced by an equivalent spring with the same stiffness, so the changes of mass center and gravitational potential energy during the compression process of the jack can be ignored. It is defined that the height between the initial position where coal gangue particles fall freely from and the position where coal gangue particles initially contact with the upper surface of the tail beam and ish, and the velocity of the particles in the initial contact is as follows:(12)v=2gh,(13)v⊥=2gh⋅sinξ.When the rock sphere impacts the tail beam and is compressed, the change of its potential energy is obtained:(14)ECS=mg⋅R⋅cosξ−Γ⋅cosξ−ψ−R⋅cosξ+γ−Γ⋅cosξ+γ−ψ+δ⋅cosξ+γ.According to the law of energy conservation of the system,(15)
1
2
m
⋅
v
⊥
2
+
E
C
S
=
∫
0
δ
F
Con
d
δ
+
2
⋅
1
2
K
Q
⋅
Δ
s
2
.On the basis of equations (4), (8), (10), (11), (13), (14),and (15), the system contact response when the rock sphere impacting the tail beam is closely related to various parameters, such as the mass and properties of the impact object, the impact height, the contact position, and the initial position of the tail beam.


## 4. Dynamic Simulation of the Impact of the Coal Gangue Particles on the Tail Beam

To study the system contact response when the tail beam is impacted by the coal gangue particles deeply, ABAQUS software will be used to analyze the dynamic simulation of the coal gangue particles impacting the tail beam based on the kinematics and dynamics models of the coal gangue particle impacting the tail beam mentioned above.


### 4.1. Simulation Model and Its Mesh Generation

The 3D model of the coal gangue particles impacting the tail beam of the hydraulic support is established in Solidworks. In order to achieve a qualitative analysis, coal gangue particles are equivalent to elastic spheres, and the radius of particles in the simulation is 25 mm other than the variable radius simulation. The rock sphere owns small mass than that of the heavy hydraulic support. As a result, the impact response of the whole hydraulic support caused by the impact-contact force of the sphere and the tail beam is small. So the gravity of the tail beam is ignored to amplify the impact-contact response. At the same time, during the establishment of the dynamic model of the coal gangue particles impacting the tail beam, the influences of the top beam, shield beam, front and rear connecting rods, the base, and insert plate are not considered. According to Figure4, the tail beam device is equivalent to the structural components where a tail beam sustained by the supporting frame. While the supporting frame supporting the tail beam replaces the shield beam and other components, the dip angle of the tail beam is not changed. And the hydraulic oil inside the jack of the tail beam exerts a great influence on the system response. However, the complexity and difficulty of the liquid simulation in ABAQUS software are high. So the jack and the hydraulic oil are replaced by the spring with the same stiffness. During the modeling process, only the terminus of the hydraulic rod of the jack and the bottom of the hydraulic cylinder is retained so as to arrange the spring, as shown in Figure 10. In order to shorten the simulation time, the free falling process of the rock sphere is not covered. And a tiny gap is reserved between the rock sphere and the metal plate in overlying strata of the tail beam during the modeling process. By applying an initial vertical velocity to the rock sphere which is similar to the initial contact velocity of the sphere obtained from the free falling, the free falling process of the rock sphere is compensated.Figure 10
Simulation model after meshed.The 3D model of the coal gangue particles impacting the tail beam is introduced into ABAQUS and is meshed. The rock sphere is freely meshed by means of the mesh generation function of ABAQUS. The mesh size is 0.005 m, totaling 3,701 meshes. Hexahedral mesh can promote the calculation accuracy. However, the size of the tail beam is large and the structure of the box body is complicated. If all the tail beams are divided into hexahedral meshes, the calculation time will be greatly increased. Therefore, the metal plate of the overlying strata of tail beam (will be impacted and contacted by the rock sphere) is divided into a hexahedral mesh, as well as the connecting position between the tail beam and the support frame. And the rest is meshed freely into the tetrahedral meshes. Firstly, the tail beam will be cut by using the partition function of the software, while the metal plate of the overlying strata of the tail beam and the connecting position between the tail beam and the supporting frame is cut into standard shapes. And they will be divided into hexahedral meshes with a mesh size of 0.02 m while the rest of the tail beam is a tetrahedral mesh with a size of 0.02 m. After the completion of the mesh generation, the tail beam has a total of 58,842 meshes. The supporting frame is featured by a simple structure and is cut into several standard shape bodies. Then, it will be divided into a hexahedral mesh with a size of 0.02 m, totaling 6,263 meshes. Furthermore, an equivalent spring will be added between the terminus of the hydraulic rod of the jack and the bottom of the hydraulic cylinder. In order to reduce the coupling calculation difficulties of the finite element mesh calculation and equivalent spring force, they are meshed freely into tetrahedral meshes and the mesh size is 0.02 m. After the completion of the mesh generation, the bottom of the hydraulic cylinder has 4,118 meshes while the terminus of hydraulic rod has 6,545 meshes. The pin of the joint between the terminus of the hydraulic rod and the tail beam is divided by the automatic division method, and the mesh size is 0.005 m, with a total of 1566 meshes.


### 4.2. Settings for Contact Response and Equivalent Spring

During the impact behavior of the coal gangue particles on the tail beam, the spherical surface will be in contact with the metal plate of the overlaying strata of the tail beam, thereby causing relative movement between connectors. Therefore, the surface contact between the sphere and the metal plate of the overlaying strata of the tail beam is set. To ensure the effect of force transmission, the meshes close to the hole and the hinge pin connected to the axle hole are set as rigid elements. In the later stage of exacting the data such as the velocity and acceleration of the tail beam, to ensure the data are not affected by the vibration of the meshes itself, the velocity and acceleration data of the rigid node (near the hinged hole connected with the jack of the tail beam) at the rear side of the tail beam are selected to analyze. The position of the selected node is shown in Figure11.Figure 11
Position of the selected rigid node on the tail beam.Under the flexible supporting of the jack (which is equivalently replaced by the spring in the simulation process), a relative rotation between the tail beam and other parts will occur after impact. Therefore, rotating pairs are added to the parts connected with the axle and the hole (between the tail beam and the supporting frame, between the terminus of the hydraulic rod of the jack and the supporting frame, and between the bottom of the hydraulic cylinder of the jack and the tail beam). The boundary condition is applied to the lower end-plate of the supporting frame, in which the constraint type is full constraint, and the constraint position is shown in Figure12(a). The gravity of the tail beam itself is larger while the volume and mass of the rock sphere are smaller. Therefore, when the gravity acceleration of −9.8 m/s2 is applied to the rock sphere, the magnitude of contact force is limited. If the jack compression caused by the gravity of the tail beam is considered, the response amplitude of the tail beam and the spring will be weakened under the contact force between the rock and the tail beam. This paper aims to study the system contact and vibration response caused by the impact of coal-rock particles on the tail beam. Therefore, the gravity of the tail beam is ignored, and the gravity acceleration is also not added.Figure 12
Settings of (a) the boundary conditions and (b) the equivalent spring.
(a)
(b)In the process of modeling, the spring is proposed to be utilized to simulate the deformation and force transmission of the jack when it is impacted. Virtual spring is used and connected at the center point between the terminus of the hydraulic rod and the bottom of the hydraulic cylinder, as shown in Figure12(b). Moreover, parameters of the spring such as the equivalent stiffness and damping coefficient are set among which the former is calculated to be 2.12253 × 108 N/m3/2 in accordance with equation (4) and Figure 13 (the structure and size diagram of the jack of the tail beam). The spring damp only changes the response time but has no effect on the response process. Therefore, the damping coefficient of the spring is set as 12,000 in order to accelerate the response rate.Figure 13
Dimension and structure diagram of the jack of the tail beam.


### 4.3. Simulation Types

Material properties have an effect on the contact stiffness and then on the system response in the process of impact contact. The height of the falling body determines the initial contact velocity, and the impact position of the tail beam affects the stress state of the impact system. Therefore, both the falling height of the coal gangue particles and the impact position of the tail beam have influences on the system response during the impact-contact process. In order to study the influences of material properties, size of the impacted object, height of the falling body, and impact position of the tail beam on the system contact response, four different simulations are carried out in ABAQUS software, respectively: the coal gangue particles impact the tail beam from different heights, the coal gangue particles impact different positions of the tail beam, coal gangue particles with different radii impact the tail beam, and the coal gangue particles with different materials impact the tail beam.


#### 4.3.1. Different Impact Heights

Take falling height of coal particles from the free falling position to the position where particles initially contact with the tail beam as 2 m, 2.5 m, 3m, 3.5 m, and 4m, respectively. The material parameters of particles, the tail beam, and other components (both are Q345) are shown in Table1.Table 1
Parameters of coal particles and Q345.
Materials
Density (kg/m3)
Young’s modulus (Pa)
Poisson’s ratio
Coal
1380
2.20e + 9
0.28
Q345
7850
2.06e + 11
0.3During the establishment of the 3D model, it has been mentioned that because of the complexity and time-consuming calculation of the model, the free falling increasing process of the particles is replaced by applying a certain initial velocity to the coal particles. The 0.008 m gap between the particles and the metal plate of the overlaying strata of the tail beam is reserved (which has little effect on the velocity). It is calculated that the initial velocities of the five groups of simulations at different heights are 6.261 m/s, 7 m/s, 7.668 m/s, 8.368 m/s, and 8.854 m/s, respectively. The lateral coordinate of the impact position is −1 m, as shown in Figure14. And the rest settings of the simulation are shown in Table 2. The simulation time of each group is 0.05 s.Figure 14
Impact position.Table 2
Partial simulation parameters at different impact heights.
Simulation group
Particle falling height (corresponding initial velocity)
X coordinate (m)
Particle material
Job-1
2 m (6.261 m/s)
−1
Coal
Job-2
2.5 m (7 m/s)
−1
Coal
Job-3
3 m (7.668 m/s)
−1
Coal
Job-4
3.5 m (8.368 m/s)
−1
Coal
Job-5
4 m (8.854 m/s)
−1
Coal


#### 4.3.2. Different Impact Positions

Take the falling height of the coal particles from the free falling position to the position where particles initially contact with the tail beam as 3.5 m. In other words, the initial velocity applied to the coal particles is 8.368 m/s. During the modeling process, the particle center is set to be corresponding to different abscissax so that the particles can impact different positions of the tail beam. The material parameters of particles, the tail , and other components (both are Q345) are shown in Table 1. The simulation time of each group is 0.05 s, and the rest settings of the simulation are shown in Table 3. Figure 15 shows the different impact positions.Table 3
Parameters at different positions.
Group
Height
x (m)
Material
Job-6
3.5 m
−1.25
Coal
Job-7
3.5 m
−1.20
Coal
Job-8
3.5 m
−1.15
Coal
Job-9
3.5 m
−1.10
Coal
Job-10
3.5 m
−1.05
Coal
Job-4
3.5 m
−1.00
Coal
Job-11
3.5 m
−0.95
Coal
Job-12
3.5 m
−0.90
Coal
Job-13
3.5 m
−0.85
Coal
Job-14
3.5 m
−0.80
Coal
Job-15
3.5 m
−0.75
CoalFigure 15
Different impact positions.


#### 4.3.3. Different Sizes of the Impact Object

Also, take the falling height of the coal particles from the free falling position to the position where particles initial contact with the tail beam as 3.5 m, i.e., the initial velocity applied to the coal particles is 8.368 m/s. The lateral coordinate of the impact position is −1 m, as shown in Figure14. The radii of the coal particles are 0.025 m, 0.030 m, 0.035 m, and 0.040 m, respectively. The material parameters are shown in Table 1, and the simulation time of each group is 0.05 s.


#### 4.3.4. Different Impact Materials

Take the falling height of the coal gangue particles from the free falling position to the position where particles initially contact with the tail beam as 3.5 m. The lateral coordinate of the impact position isX = −1 m, as shown in Figure 14. Coal gangue materials are taken from the coal, the sandstone, and marble, and the material parameters are shown in Table 4. Table 4 also shows the material parameters of the tail beam and other components (all are Q345). The simulation time of each group is 0.05 s.Table 4
Material parameters of coal gangue and Q345.
Material
Density (kg/m3)
Young’s modulus (Pa)
Poisson’s ratio
Coal
1380
2.20e + 9
0.28
Sandstone
2487
1.35e + 10
0.123
Marble
2800
3.40e + 10
0.3
Q345
7850
2.06e + 11
0.3


## 4.1. Simulation Model and Its Mesh Generation

The 3D model of the coal gangue particles impacting the tail beam of the hydraulic support is established in Solidworks. In order to achieve a qualitative analysis, coal gangue particles are equivalent to elastic spheres, and the radius of particles in the simulation is 25 mm other than the variable radius simulation. The rock sphere owns small mass than that of the heavy hydraulic support. As a result, the impact response of the whole hydraulic support caused by the impact-contact force of the sphere and the tail beam is small. So the gravity of the tail beam is ignored to amplify the impact-contact response. At the same time, during the establishment of the dynamic model of the coal gangue particles impacting the tail beam, the influences of the top beam, shield beam, front and rear connecting rods, the base, and insert plate are not considered. According to Figure4, the tail beam device is equivalent to the structural components where a tail beam sustained by the supporting frame. While the supporting frame supporting the tail beam replaces the shield beam and other components, the dip angle of the tail beam is not changed. And the hydraulic oil inside the jack of the tail beam exerts a great influence on the system response. However, the complexity and difficulty of the liquid simulation in ABAQUS software are high. So the jack and the hydraulic oil are replaced by the spring with the same stiffness. During the modeling process, only the terminus of the hydraulic rod of the jack and the bottom of the hydraulic cylinder is retained so as to arrange the spring, as shown in Figure 10. In order to shorten the simulation time, the free falling process of the rock sphere is not covered. And a tiny gap is reserved between the rock sphere and the metal plate in overlying strata of the tail beam during the modeling process. By applying an initial vertical velocity to the rock sphere which is similar to the initial contact velocity of the sphere obtained from the free falling, the free falling process of the rock sphere is compensated.Figure 10
Simulation model after meshed.The 3D model of the coal gangue particles impacting the tail beam is introduced into ABAQUS and is meshed. The rock sphere is freely meshed by means of the mesh generation function of ABAQUS. The mesh size is 0.005 m, totaling 3,701 meshes. Hexahedral mesh can promote the calculation accuracy. However, the size of the tail beam is large and the structure of the box body is complicated. If all the tail beams are divided into hexahedral meshes, the calculation time will be greatly increased. Therefore, the metal plate of the overlying strata of tail beam (will be impacted and contacted by the rock sphere) is divided into a hexahedral mesh, as well as the connecting position between the tail beam and the support frame. And the rest is meshed freely into the tetrahedral meshes. Firstly, the tail beam will be cut by using the partition function of the software, while the metal plate of the overlying strata of the tail beam and the connecting position between the tail beam and the supporting frame is cut into standard shapes. And they will be divided into hexahedral meshes with a mesh size of 0.02 m while the rest of the tail beam is a tetrahedral mesh with a size of 0.02 m. After the completion of the mesh generation, the tail beam has a total of 58,842 meshes. The supporting frame is featured by a simple structure and is cut into several standard shape bodies. Then, it will be divided into a hexahedral mesh with a size of 0.02 m, totaling 6,263 meshes. Furthermore, an equivalent spring will be added between the terminus of the hydraulic rod of the jack and the bottom of the hydraulic cylinder. In order to reduce the coupling calculation difficulties of the finite element mesh calculation and equivalent spring force, they are meshed freely into tetrahedral meshes and the mesh size is 0.02 m. After the completion of the mesh generation, the bottom of the hydraulic cylinder has 4,118 meshes while the terminus of hydraulic rod has 6,545 meshes. The pin of the joint between the terminus of the hydraulic rod and the tail beam is divided by the automatic division method, and the mesh size is 0.005 m, with a total of 1566 meshes.


## 4.2. Settings for Contact Response and Equivalent Spring

During the impact behavior of the coal gangue particles on the tail beam, the spherical surface will be in contact with the metal plate of the overlaying strata of the tail beam, thereby causing relative movement between connectors. Therefore, the surface contact between the sphere and the metal plate of the overlaying strata of the tail beam is set. To ensure the effect of force transmission, the meshes close to the hole and the hinge pin connected to the axle hole are set as rigid elements. In the later stage of exacting the data such as the velocity and acceleration of the tail beam, to ensure the data are not affected by the vibration of the meshes itself, the velocity and acceleration data of the rigid node (near the hinged hole connected with the jack of the tail beam) at the rear side of the tail beam are selected to analyze. The position of the selected node is shown in Figure11.Figure 11
Position of the selected rigid node on the tail beam.Under the flexible supporting of the jack (which is equivalently replaced by the spring in the simulation process), a relative rotation between the tail beam and other parts will occur after impact. Therefore, rotating pairs are added to the parts connected with the axle and the hole (between the tail beam and the supporting frame, between the terminus of the hydraulic rod of the jack and the supporting frame, and between the bottom of the hydraulic cylinder of the jack and the tail beam). The boundary condition is applied to the lower end-plate of the supporting frame, in which the constraint type is full constraint, and the constraint position is shown in Figure12(a). The gravity of the tail beam itself is larger while the volume and mass of the rock sphere are smaller. Therefore, when the gravity acceleration of −9.8 m/s2 is applied to the rock sphere, the magnitude of contact force is limited. If the jack compression caused by the gravity of the tail beam is considered, the response amplitude of the tail beam and the spring will be weakened under the contact force between the rock and the tail beam. This paper aims to study the system contact and vibration response caused by the impact of coal-rock particles on the tail beam. Therefore, the gravity of the tail beam is ignored, and the gravity acceleration is also not added.Figure 12
Settings of (a) the boundary conditions and (b) the equivalent spring.
(a)
(b)In the process of modeling, the spring is proposed to be utilized to simulate the deformation and force transmission of the jack when it is impacted. Virtual spring is used and connected at the center point between the terminus of the hydraulic rod and the bottom of the hydraulic cylinder, as shown in Figure12(b). Moreover, parameters of the spring such as the equivalent stiffness and damping coefficient are set among which the former is calculated to be 2.12253 × 108 N/m3/2 in accordance with equation (4) and Figure 13 (the structure and size diagram of the jack of the tail beam). The spring damp only changes the response time but has no effect on the response process. Therefore, the damping coefficient of the spring is set as 12,000 in order to accelerate the response rate.Figure 13
Dimension and structure diagram of the jack of the tail beam.


## 4.3. Simulation Types

Material properties have an effect on the contact stiffness and then on the system response in the process of impact contact. The height of the falling body determines the initial contact velocity, and the impact position of the tail beam affects the stress state of the impact system. Therefore, both the falling height of the coal gangue particles and the impact position of the tail beam have influences on the system response during the impact-contact process. In order to study the influences of material properties, size of the impacted object, height of the falling body, and impact position of the tail beam on the system contact response, four different simulations are carried out in ABAQUS software, respectively: the coal gangue particles impact the tail beam from different heights, the coal gangue particles impact different positions of the tail beam, coal gangue particles with different radii impact the tail beam, and the coal gangue particles with different materials impact the tail beam.


### 4.3.1. Different Impact Heights

Take falling height of coal particles from the free falling position to the position where particles initially contact with the tail beam as 2 m, 2.5 m, 3m, 3.5 m, and 4m, respectively. The material parameters of particles, the tail beam, and other components (both are Q345) are shown in Table1.Table 1
Parameters of coal particles and Q345.
Materials
Density (kg/m3)
Young’s modulus (Pa)
Poisson’s ratio
Coal
1380
2.20e + 9
0.28
Q345
7850
2.06e + 11
0.3During the establishment of the 3D model, it has been mentioned that because of the complexity and time-consuming calculation of the model, the free falling increasing process of the particles is replaced by applying a certain initial velocity to the coal particles. The 0.008 m gap between the particles and the metal plate of the overlaying strata of the tail beam is reserved (which has little effect on the velocity). It is calculated that the initial velocities of the five groups of simulations at different heights are 6.261 m/s, 7 m/s, 7.668 m/s, 8.368 m/s, and 8.854 m/s, respectively. The lateral coordinate of the impact position is −1 m, as shown in Figure14. And the rest settings of the simulation are shown in Table 2. The simulation time of each group is 0.05 s.Figure 14
Impact position.Table 2
Partial simulation parameters at different impact heights.
Simulation group
Particle falling height (corresponding initial velocity)
X coordinate (m)
Particle material
Job-1
2 m (6.261 m/s)
−1
Coal
Job-2
2.5 m (7 m/s)
−1
Coal
Job-3
3 m (7.668 m/s)
−1
Coal
Job-4
3.5 m (8.368 m/s)
−1
Coal
Job-5
4 m (8.854 m/s)
−1
Coal


### 4.3.2. Different Impact Positions

Take the falling height of the coal particles from the free falling position to the position where particles initially contact with the tail beam as 3.5 m. In other words, the initial velocity applied to the coal particles is 8.368 m/s. During the modeling process, the particle center is set to be corresponding to different abscissax so that the particles can impact different positions of the tail beam. The material parameters of particles, the tail , and other components (both are Q345) are shown in Table 1. The simulation time of each group is 0.05 s, and the rest settings of the simulation are shown in Table 3. Figure 15 shows the different impact positions.Table 3
Parameters at different positions.
Group
Height
x (m)
Material
Job-6
3.5 m
−1.25
Coal
Job-7
3.5 m
−1.20
Coal
Job-8
3.5 m
−1.15
Coal
Job-9
3.5 m
−1.10
Coal
Job-10
3.5 m
−1.05
Coal
Job-4
3.5 m
−1.00
Coal
Job-11
3.5 m
−0.95
Coal
Job-12
3.5 m
−0.90
Coal
Job-13
3.5 m
−0.85
Coal
Job-14
3.5 m
−0.80
Coal
Job-15
3.5 m
−0.75
CoalFigure 15
Different impact positions.


### 4.3.3. Different Sizes of the Impact Object

Also, take the falling height of the coal particles from the free falling position to the position where particles initial contact with the tail beam as 3.5 m, i.e., the initial velocity applied to the coal particles is 8.368 m/s. The lateral coordinate of the impact position is −1 m, as shown in Figure14. The radii of the coal particles are 0.025 m, 0.030 m, 0.035 m, and 0.040 m, respectively. The material parameters are shown in Table 1, and the simulation time of each group is 0.05 s.


### 4.3.4. Different Impact Materials

Take the falling height of the coal gangue particles from the free falling position to the position where particles initially contact with the tail beam as 3.5 m. The lateral coordinate of the impact position isX = −1 m, as shown in Figure 14. Coal gangue materials are taken from the coal, the sandstone, and marble, and the material parameters are shown in Table 4. Table 4 also shows the material parameters of the tail beam and other components (all are Q345). The simulation time of each group is 0.05 s.Table 4
Material parameters of coal gangue and Q345.
Material
Density (kg/m3)
Young’s modulus (Pa)
Poisson’s ratio
Coal
1380
2.20e + 9
0.28
Sandstone
2487
1.35e + 10
0.123
Marble
2800
3.40e + 10
0.3
Q345
7850
2.06e + 11
0.3


## 4.3.1. Different Impact Heights

Take falling height of coal particles from the free falling position to the position where particles initially contact with the tail beam as 2 m, 2.5 m, 3m, 3.5 m, and 4m, respectively. The material parameters of particles, the tail beam, and other components (both are Q345) are shown in Table1.Table 1
Parameters of coal particles and Q345.
Materials
Density (kg/m3)
Young’s modulus (Pa)
Poisson’s ratio
Coal
1380
2.20e + 9
0.28
Q345
7850
2.06e + 11
0.3During the establishment of the 3D model, it has been mentioned that because of the complexity and time-consuming calculation of the model, the free falling increasing process of the particles is replaced by applying a certain initial velocity to the coal particles. The 0.008 m gap between the particles and the metal plate of the overlaying strata of the tail beam is reserved (which has little effect on the velocity). It is calculated that the initial velocities of the five groups of simulations at different heights are 6.261 m/s, 7 m/s, 7.668 m/s, 8.368 m/s, and 8.854 m/s, respectively. The lateral coordinate of the impact position is −1 m, as shown in Figure14. And the rest settings of the simulation are shown in Table 2. The simulation time of each group is 0.05 s.Figure 14
Impact position.Table 2
Partial simulation parameters at different impact heights.
Simulation group
Particle falling height (corresponding initial velocity)
X coordinate (m)
Particle material
Job-1
2 m (6.261 m/s)
−1
Coal
Job-2
2.5 m (7 m/s)
−1
Coal
Job-3
3 m (7.668 m/s)
−1
Coal
Job-4
3.5 m (8.368 m/s)
−1
Coal
Job-5
4 m (8.854 m/s)
−1
Coal


## 4.3.2. Different Impact Positions

Take the falling height of the coal particles from the free falling position to the position where particles initially contact with the tail beam as 3.5 m. In other words, the initial velocity applied to the coal particles is 8.368 m/s. During the modeling process, the particle center is set to be corresponding to different abscissax so that the particles can impact different positions of the tail beam. The material parameters of particles, the tail , and other components (both are Q345) are shown in Table 1. The simulation time of each group is 0.05 s, and the rest settings of the simulation are shown in Table 3. Figure 15 shows the different impact positions.Table 3
Parameters at different positions.
Group
Height
x (m)
Material
Job-6
3.5 m
−1.25
Coal
Job-7
3.5 m
−1.20
Coal
Job-8
3.5 m
−1.15
Coal
Job-9
3.5 m
−1.10
Coal
Job-10
3.5 m
−1.05
Coal
Job-4
3.5 m
−1.00
Coal
Job-11
3.5 m
−0.95
Coal
Job-12
3.5 m
−0.90
Coal
Job-13
3.5 m
−0.85
Coal
Job-14
3.5 m
−0.80
Coal
Job-15
3.5 m
−0.75
CoalFigure 15
Different impact positions.


## 4.3.3. Different Sizes of the Impact Object

Also, take the falling height of the coal particles from the free falling position to the position where particles initial contact with the tail beam as 3.5 m, i.e., the initial velocity applied to the coal particles is 8.368 m/s. The lateral coordinate of the impact position is −1 m, as shown in Figure14. The radii of the coal particles are 0.025 m, 0.030 m, 0.035 m, and 0.040 m, respectively. The material parameters are shown in Table 1, and the simulation time of each group is 0.05 s.


## 4.3.4. Different Impact Materials

Take the falling height of the coal gangue particles from the free falling position to the position where particles initially contact with the tail beam as 3.5 m. The lateral coordinate of the impact position isX = −1 m, as shown in Figure 14. Coal gangue materials are taken from the coal, the sandstone, and marble, and the material parameters are shown in Table 4. Table 4 also shows the material parameters of the tail beam and other components (all are Q345). The simulation time of each group is 0.05 s.Table 4
Material parameters of coal gangue and Q345.
Material
Density (kg/m3)
Young’s modulus (Pa)
Poisson’s ratio
Coal
1380
2.20e + 9
0.28
Sandstone
2487
1.35e + 10
0.123
Marble
2800
3.40e + 10
0.3
Q345
7850
2.06e + 11
0.3


## 5. Results, Discussion, and Analysis

### 5.1. Contact Response of the System with Different Impact Heights

Coal-like rock sphere shows different impact velocities and impact energy when it impacts the tail beam from different heights. Under various impact heights, the curves of the spring stress, the system contact force, the acceleration and velocity of the rigid element close to the hinged hole of the tail beam, and the acceleration and velocity of the coal-like rock sphere are shown in Figures16–21. In this paper, the acceleration and velocity of the rock sphere extracted from the simulation refer to the corresponding data of the center node of the rock sphere. Due to the violent vibration of the selected rigid node of the tail beam, the acceleration curve of the rigid node close to the hinged hole of the tail beam is processed by FFT for standard analysis.Figure 16
Spring stress.Figure 17
Contact force.Figure 18
Acceleration of the tail beam.Figure 19
Velocity of the tail beam.Figure 20
Acceleration of the sphere.Figure 21
Velocity of the sphere.As shown Figures16–21, as the simulation time changes and under different impact velocities (i.e., different impact heights), the form of curves of the spring stress, the contact force between the rock sphere and the tail beam, the vibration velocity of the selected rigid node of tail beam, and the vibration velocity and acceleration of the rock sphere are all the same. And they all grow as the impact velocity increases. The reason is that when the rock sphere impacts the tail beam from different heights, the higher the height is, the stronger the impact velocity is. Furthermore, the initial impact energy in the impact-contact system composed of the rock sphere and the tail beam is larger, which leads to increases in above contact responses. The contact force only occurs in the very short time when the rock sphere impacts the tail beam. After the impact, it disappears and the reciprocating vibration of the spring occurs, which causes the tail beam to reciprocate and vibrate. Under the action of the spring, the vibration velocity of the selected rigid node presents an approximately periodic reciprocating fluctuation that increases firstly and decreases secondly. In addition, the vibration amplitude of the tail beam gradually declines. After the impact, the velocity and acceleration curves of the rock sphere quickly remain steady.The maximum spring stress (which is defined as the absolute value of the minimum negative value of the spring stress curve) at different impact heights, the maximum system contact force, the maximum acceleration (after FFT treatment) and velocity of the selected rigid node of the tail beam, and the maximum acceleration and velocity of the coal-like rock sphere are extracted. As a result, the maximum contact response of the system at different impact height is obtained, as shown in Figure22. As the impact height rises, the maximum spring stress, the maximum system contact force, and the maximum acceleration and velocity of the coal-like rock sphere gradually increase while their growth rates decrease. The acceleration of the rigid node close to the hinged hole of the tail beam increases in linear approximation, and its velocity as well as the growth rate increases gradually. It can be seen that the greater the impact height is, the more obvious the vibration amplitude of the tail beam is. The vibration amplitude of the tail beam varies sharply with the increase of impact height. The maximum rebound velocities of five rock spheres is 4.23424 m/s, 4.73325 m/s, 5.18517 m/s, 5.65751 m/s, and 5.98562 m/s, respectively. And their corresponding impact velocities are 6.261 m/s, 7 m/s, 7.668 m/s, 8.368 m/s, and 8.584 m/s. The comprehensive recovery coefficients (defined as the ratio of the total rebound velocity to the initial contact velocity) of the five groups of rock spheres with different impact heights are 0.676288132, 0.6761781, 0.67620892, 0.676088671, and 0.697299627, respectively. The variation of the comprehensive recovery coefficient at different impact heights is very small.Figure 22
Maximum contact response.


### 5.2. Contact Response of the System with Different Impact Positions

The tail beam is sustained by the supporting frame and the jack of the tail beam. Under the impact-contact force of the coal-like rock sphere, a four-in-one impact response system composed by the rock sphere, the tail beam, the jack of the tail beam, and the supporting frame is formed. If the rock sphere impacts the different positions of the tail beam, the distance between the contact position of the tail beam and the rigid support terminus of the tail beam (the support terminus by the supporting frame) is different, as well as between the contact position and the elastic support terminus (the support terminus by the jack). As a result, structural stress of the impact system changes and then affects the system impact-contact response. When the rock sphere impacts different positions of the axis of the tail beam at the same velocity, the curves of the spring stress, the system contact force, the acceleration and velocity of the selected rigid node of the tail beam, and the acceleration and velocity of the coal-like rock are shown in Figures23–28. In addition, the acceleration curve of the rigid element close to the hinged hole of tail beam is also treated by FFT.Figure 23
Spring stress.Figure 24
Contact force.Figure 25
Acceleration of the tail beam.Figure 26
Velocity of the tail beam.Figure 27
Acceleration of the sphere.Figure 28
Velocity of the sphere.It can be seen from the Figures23–28 that the curves of the system contact response are different when the coal-like rock spheres with the same radius impact different positions of the tail beams at the same velocity, and they does not change monotonically with the change of the impact position. The velocity and acceleration of the rock sphere are approximately the same during the impact process, which indicates that the residual energy of the rock sphere is almost constant when the rock sphere impacts the different positions of the tail beam, and the total amount of the energy transferred to the tail beam and the spring from the rock sphere is approximately the same. In order to analyze the law of system response further, the maximum spring stress (which is defined as the absolute value of the minimum negative value of the spring stress curve) at different impact heights, the maximum system contact force, the maximum acceleration (after FFT treatment) and velocity of the selected rigid node of the tail beam, and the maximum acceleration and velocity of the coal-like rock sphere are extracted. Figure 29 shows the maximum system contact response.Figure 29
Maximum contact response.Figure29 shows that as the impact position changes, the maximum value of the maximum spring stress appears when the rock sphere impacts the tail beam near the support terminus of the spring. Maximum spring stress decreases first and then increases when the impact position moves away from the support terminus of the spring. And when it comes near to x = −1 m, the maximum value of the maximum spring stress appears again. When the impacted position continues to approach the rigid support terminus, the maximum spring stress decreases again and it increases slightly near the terminus. The trend of the maximum spring stress of the system, as a whole, appears decrease ⟶ increase ⟶ decrease ⟶ increase ⟶ decrease ⟶ increase (the direction is from the support terminus of the spring to the rigid support terminus, i.e., x = −1.3 m ⟶ x = −0.75 m). And the minimum value of the maximum spring stress of the system appears around x = −1.25 m. The maximum contact force of the system, as a whole, shows an increase ⟶ decrease ⟶ increase ⟶ decrease ⟶ increase trend. The maximum contact force reaches its maximum value around x = −0.95 m while its minimum value appears near x = −1.2 m. That is, the minimum value of the maximum contact force occurs at a short distance from the support terminus of the spring, and the minimum value of the maximum contact force appears close to the center of the tail beam but slightly biased toward the fixed support terminus. The maximum acceleration belonging to the selected rigid node of the tail beam shows a decrease ⟶ increase ⟶ decrease ⟶ increase trend (the oscillation changes after x > −1.2 m is considered to be decreasing all the time). The maximum value of the maximum acceleration of the selected rigid node occurs at x = −1.3 m that is closest to the support terminus of the spring. And the minimum value of the maximum acceleration occurs at x = −0.75 m which is closest to the rigid support terminus. The maximum velocity of the selected rigid node shows the trend of decreasing ⟶ increasing ⟶ decreasing ⟶ increasing ⟶ decreasing. The maximum value of the maximum velocity of the selected rigid node reaches its maximum value at x = −1.3 m which is closest to the support terminus of the spring while it reaches its minimum value at x = −1.15 m which is close to the center of the tail beam but slightly biased toward the elastic support terminus (spring). Therefore, the most obvious vibration signal of the tail beam occurs when the support terminus of the spring is impacted. The minimum and maximum values of the maximum acceleration of the rock sphere are 976307 m/s2 and 976474 m/s2, respectively. And the minimum and maximum values of the maximum velocity are 5.6572 m/s and 5.65799 m/s, respectively. In addition, the maximum changing rates of the maximum acceleration and velocity are 0.017105% and 0.013965%, respectively, which can be ignored. So, when the same rock sphere impacts the different positions of the tail beam along the central axis, its comprehensive recovery coefficient hardly changes.


### 5.3. Contact Response of the System with Different Impact Sizes

For impacted objects with the same properties, its different sizes will result in different influences to the contact stiffness, the impact energy, the equivalent contact radius, and the contact area of the system. Moreover, the compression depth of the rock sphere, the contact force, and the structural force of the system are also affected. Therefore, the contact response of the system will be different when the radius of the impacted object is various. As the varisized coal-like rock spheres impact the same position (x = −1 m) of the tail beam at the same velocity of 8.368 m/s (the same impact height), the curves of the spring stress, the system contact force, the acceleration and velocity of the rigid node of the tail beam, and the acceleration and velocity of the coal-like rock sphere are shown in Figures 30–35. And the acceleration curve of the rigid node close to the hinged holes of the tail beam (Figure 32) is processed by FFT.Figure 30
Spring stress.Figure 31
Contact force.Figure 32
Acceleration of the tail beam.Figure 33
Velocity of the tail beam.Figure 34
Acceleration of the sphere.Figure 35
Velocity of the sphere.As shown in Figures30–35, when the coal-like rock spheres with the different radius impact the same position of the tail beam at the same velocity, the curve form of the spring stress, the vibration velocity of the selected rigid node of the tail beam and the velocity and acceleration of the rock sphere is the same as the simulation time varies. As the radius of the coal-like rock sphere increases, the spring stress, the contact force between the rock sphere, and the tail beam as well as the vibration velocity and acceleration of the selected rigid node of the tail beam increase, while the velocity and acceleration of the rock sphere decrease. The reason is that the material properties remain constant, the larger the radius of the rock sphere is, the greater the Hertz contact stiffness of the system is. Furthermore, the mass and the initial impact energy of the coal-like rock sphere also increase. As a result, the contact force between the rock sphere and the tail beam increases and more energy is transferred to the tail beam when it is impacted. Therefore, the vibration velocity and acceleration of the rigid node of the tail beam increase. The energy transferred to the tail beam increases, resulting in that the ratio of the energy of the coal-like rock sphere to its initial impact energy decreases after impact contact. Furthermore, after the impact-contact process, the larger the radius of the rock sphere is, the smaller the velocity and acceleration of the rock sphere are. Similarly, the velocity and acceleration curves of the rock sphere quickly remain steady.The same treatment is also utilized to, respectively, extract the maximum values of the contact responses in Figures30–35 when the rock spheres with different radii impact the tail beams (where the maximum acceleration of the selected rigid node is equal to the maximum value of the acceleration curve after FFT treatment, and the maximum spring stress is defined as the absolute value of the minimum negative value of the spring stress curve). Then, the maximum contact response of the system at different impact heights is obtained, as shown in Figure 36. According to Figure 36, it can be seen that as the radius of the rock sphere increases, the maximum spring stress, the maximum system contact force, and the velocity of the rigid element close to the hinged hole of the tail beam increase gradually, and their growth rates increase. This shows that the larger the radius of the rock sphere is, the more obvious the vibration amplitude of the tail beam is. And the acceleration of the rigid node close to the hinged hole of the tail beam increases gradually while its growth rate decreases slightly. As the radius of the rock sphere increase, its maximum velocity and acceleration decrease and the decreasing rate gradually slows down. The impact velocities of rock spheres are 8.368 m/s, and the maximum rebound velocities of four groups of rock spheres with different radii are 5.65751 m/s, 3.96624 m/s, 3.57084 m/s, and 2.78333 m/s, respectively. And their comprehensive recovery coefficients are 0.676088671, 0.473977055, 0.426725621, and 0.332615917, respectively. When the rock sphere impacts the tail beam, the comprehensive recovery coefficient of the rock sphere decreases as its radius increases.Figure 36
Maximum contact response.


### 5.4. Contact Response of the System with Different Impact Materials

Two kinds of particles coal and gangue can be seen in the process of coal mining. And the properties of coal and gangue under different working faces are also different. The contact stiffness and the damping of the system vary with the material of the contacted object, which leads to the change of the system impact-contact response when the particles impact the tail beam. In this paper, one kind of coal and two different kinds of gangue (sandstone and marble) are considered as the research object, as shown in Table4. When the different elastic spheres (which are defined as coal elastic sphere, sandstone elastic sphere, and marble elastic sphere) with the same radius, respectively, impact the same position (x = −1 m) of the tail beam at the same velocity (the same impact height) of 8.368 m/s, the curves of the spring stress, the system contact force, the acceleration and velocity of the selected rigid node of the tail beam, and the acceleration and velocity of the coal-rock sphere are shown in Figures 37–42.Figure 37
Spring stress.Figure 38
Contact force.Figure 39
Acceleration of the tail beam.Figure 40
Velocity of the tail beam.Figure 41
Acceleration of the sphere.Figure 42
Velocity of the sphere.Figures37–42 show that when the coal elastic sphere, sandstone elastic sphere, and marble elastic sphere with the same radius impact the tail beam at the same velocity, the curve forms of the spring stress, the vibration acceleration and velocity of the selected rigid node of the tail beam and the vibration acceleration and velocity of the rock sphere are the same as the simulation time changes. When the same position of the tail beam is impacted at the same velocity, the spring stresses, and the vibration velocity of the rigid node of the tail beam produced by the impacts of sandstone and marble elastic spheres with the same radius are almost the same, the impacts of the sandstone elastic sphere are slightly smaller than those of the marble elastic sphere. When the coal-like rock sphere impacts the tail beam, the spring stress, the vibration acceleration and velocity of the selected rigid node of the tail beam, and the acceleration of the rock sphere are far less than those of the sandstone elastic sphere and the marble elastic sphere. However, the velocity law of the rock sphere is contrary (it means that when the rock sphere with the same radius impacts the same position of tail beam at the same velocity, the velocity of the coal elastic sphere is much faster than that of the sandstone elastic sphere and marble elastic sphere).Figures37–42 also show that when the coal elastic sphere impacts the tail beam, the maximum single spring stress is 33.685 MPa and the minimum single spring stress is −49.8086 MPa. And the maximum changing value of the single spring stress is 83.4936 MPa. Another, when the sandstone elastic sphere impacts the tail beam, the maximum single spring stress is 55.4426 MPa and the minimum single spring stress is −82.6176 MPa, and the maximum changing value of single spring stress is 138.0602 MPa. And when the marble elastic sphere impacts the tail beam, the maximum single spring stress is 56.7875 MPa and the minimum single spring stress is −86.0464 MPa. Its maximum changing value of the single spring stress is 142.8339 MPa. According to the data, the maximum spring stresses (which is defined as the absolute value of the minimum negative value of the spring stress curve) of the spring caused by the impacts of the sandstone and marble elastic spheres on the tail beam are 1.6587 times and 1.72754 times of the maximum spring stress caused by the impact of coal elastic sphere. And when the sandstone and marble elastic spheres impact the tail beam, their maximum changes of single spring stresses are 1.65354 times and 1.71072 times of the maximum spring stress caused by the impact of coal elastic sphere. When the gangue-like elastic sphere impacts the tail beam, the spring stress and the maximum changing value of the single spring stress are much greater than those of the impact caused by coal-like elastic sphere. At the same time, it can be known that when the coal, sandstone, and marble elastic spheres impact the tail beam, the maximum contact forces of the system are 1450.72 N, 4249.05 N, and 6341.25, respectively. And the contact forces produced by the impacts of sandstone and marble elastic spheres are 2.92892 times and 4.3711 times that of the coal elastic sphere. The maximum acceleration of the selected rigid node of the tail beams is 116.073 m/s2, 199.969 m/s2, and 264.017 m/s2, respectively. When the sandstone and marble elastic spheres impact the tail beam, their maximum acceleration of the selected rigid node is 1.72279 times and 2.27458 times those of the coal elastic sphere. Velocities of the selected rigid node are 0.000718687 m/s, 0.00122 m/s, and 0.00137 m/s, respectively. And the maximum velocities of the selected rigid node when the sandstone and marble elastic spheres impact are 1.69754 times and 1.90625 times that of the coal elastic sphere. The maximum acceleration of the rock spheres is 976369 m/s2, 1986110 m/s2, and 1140600 m/s2. In addition, the maximum rebound accelerations of the sandstone and marble elastic spheres after their impacts are 2.03418 times and 1.16821 times that of coal elastic sphere. The maximum velocities of the three rock spheres are 5.65751 m/s, 2.72292 m/s, and 3.04507 m/s, respectively. And rebound velocities of sandstone elastic sphere and marble elastic sphere after their impacts are 0.4812 times and 0.53824 times that of coal elastic sphere.According to the above data, the contact response of coal elastic sphere impacting the tail beam, such as spring stress, system contact force, the acceleration and velocity of the selected rigid node of the tail beam, and the acceleration and velocity of the rock sphere are different from those of the impact of gangue elastic sphere. In the simulation model, the spring replaces the jack of the tail beam in the solid model. On the basis of this, when the gangue elastic sphere, respectively, impacts the tail beam, the oil pressure, its variable quantity of the jack, and the vibration velocity of the tail beam after its impact are much larger than the change of the response parameters caused by the impact of coal elastic sphere, while the rebound velocity of the gangue elastic sphere particles is much smaller than that of the coal elastic sphere after the impact. Therefore, in the process of technical researches on the coal gangue identification for longwall top coal caving or on the coal gangue identification of particles impacting the tail beam of the hydraulic support, the oil pressure of the jack of the tail beam after impact can be extracted by the pressure sensor and its changing value can be monitored. The vibration response of the tail beam after impact can be extracted by the velocity or the acceleration sensor, and the rebound velocity of the rock sphere can be pictured by using instruments such as high-speed camera. It is feasible to carry out the gangue identification in accordance with the differences of the measured parameters (note: for a qualitative research, both the coal and gangue particles are equivalent to elastomers in the simulation. Actually, they are brittle objects and coal is easier to break than gangue particles. In actual impact experiment, when the coal and gangue particles impact the tail beam, the differences of the oil pressure and its variable quantity of the jack of tail beam, as well as the vibration velocity of the tail beam after impact will be greater. Based on these parameters, the realizability to perform the coal gangue identification will be higher. However, because of the fact that coal is easier to break than gangue particles, the rebound velocity of coal may be smaller than that of gangue in the actual experiment.)


## 5.1. Contact Response of the System with Different Impact Heights

Coal-like rock sphere shows different impact velocities and impact energy when it impacts the tail beam from different heights. Under various impact heights, the curves of the spring stress, the system contact force, the acceleration and velocity of the rigid element close to the hinged hole of the tail beam, and the acceleration and velocity of the coal-like rock sphere are shown in Figures16–21. In this paper, the acceleration and velocity of the rock sphere extracted from the simulation refer to the corresponding data of the center node of the rock sphere. Due to the violent vibration of the selected rigid node of the tail beam, the acceleration curve of the rigid node close to the hinged hole of the tail beam is processed by FFT for standard analysis.Figure 16
Spring stress.Figure 17
Contact force.Figure 18
Acceleration of the tail beam.Figure 19
Velocity of the tail beam.Figure 20
Acceleration of the sphere.Figure 21
Velocity of the sphere.As shown Figures16–21, as the simulation time changes and under different impact velocities (i.e., different impact heights), the form of curves of the spring stress, the contact force between the rock sphere and the tail beam, the vibration velocity of the selected rigid node of tail beam, and the vibration velocity and acceleration of the rock sphere are all the same. And they all grow as the impact velocity increases. The reason is that when the rock sphere impacts the tail beam from different heights, the higher the height is, the stronger the impact velocity is. Furthermore, the initial impact energy in the impact-contact system composed of the rock sphere and the tail beam is larger, which leads to increases in above contact responses. The contact force only occurs in the very short time when the rock sphere impacts the tail beam. After the impact, it disappears and the reciprocating vibration of the spring occurs, which causes the tail beam to reciprocate and vibrate. Under the action of the spring, the vibration velocity of the selected rigid node presents an approximately periodic reciprocating fluctuation that increases firstly and decreases secondly. In addition, the vibration amplitude of the tail beam gradually declines. After the impact, the velocity and acceleration curves of the rock sphere quickly remain steady.The maximum spring stress (which is defined as the absolute value of the minimum negative value of the spring stress curve) at different impact heights, the maximum system contact force, the maximum acceleration (after FFT treatment) and velocity of the selected rigid node of the tail beam, and the maximum acceleration and velocity of the coal-like rock sphere are extracted. As a result, the maximum contact response of the system at different impact height is obtained, as shown in Figure22. As the impact height rises, the maximum spring stress, the maximum system contact force, and the maximum acceleration and velocity of the coal-like rock sphere gradually increase while their growth rates decrease. The acceleration of the rigid node close to the hinged hole of the tail beam increases in linear approximation, and its velocity as well as the growth rate increases gradually. It can be seen that the greater the impact height is, the more obvious the vibration amplitude of the tail beam is. The vibration amplitude of the tail beam varies sharply with the increase of impact height. The maximum rebound velocities of five rock spheres is 4.23424 m/s, 4.73325 m/s, 5.18517 m/s, 5.65751 m/s, and 5.98562 m/s, respectively. And their corresponding impact velocities are 6.261 m/s, 7 m/s, 7.668 m/s, 8.368 m/s, and 8.584 m/s. The comprehensive recovery coefficients (defined as the ratio of the total rebound velocity to the initial contact velocity) of the five groups of rock spheres with different impact heights are 0.676288132, 0.6761781, 0.67620892, 0.676088671, and 0.697299627, respectively. The variation of the comprehensive recovery coefficient at different impact heights is very small.Figure 22
Maximum contact response.


## 5.2. Contact Response of the System with Different Impact Positions

The tail beam is sustained by the supporting frame and the jack of the tail beam. Under the impact-contact force of the coal-like rock sphere, a four-in-one impact response system composed by the rock sphere, the tail beam, the jack of the tail beam, and the supporting frame is formed. If the rock sphere impacts the different positions of the tail beam, the distance between the contact position of the tail beam and the rigid support terminus of the tail beam (the support terminus by the supporting frame) is different, as well as between the contact position and the elastic support terminus (the support terminus by the jack). As a result, structural stress of the impact system changes and then affects the system impact-contact response. When the rock sphere impacts different positions of the axis of the tail beam at the same velocity, the curves of the spring stress, the system contact force, the acceleration and velocity of the selected rigid node of the tail beam, and the acceleration and velocity of the coal-like rock are shown in Figures23–28. In addition, the acceleration curve of the rigid element close to the hinged hole of tail beam is also treated by FFT.Figure 23
Spring stress.Figure 24
Contact force.Figure 25
Acceleration of the tail beam.Figure 26
Velocity of the tail beam.Figure 27
Acceleration of the sphere.Figure 28
Velocity of the sphere.It can be seen from the Figures23–28 that the curves of the system contact response are different when the coal-like rock spheres with the same radius impact different positions of the tail beams at the same velocity, and they does not change monotonically with the change of the impact position. The velocity and acceleration of the rock sphere are approximately the same during the impact process, which indicates that the residual energy of the rock sphere is almost constant when the rock sphere impacts the different positions of the tail beam, and the total amount of the energy transferred to the tail beam and the spring from the rock sphere is approximately the same. In order to analyze the law of system response further, the maximum spring stress (which is defined as the absolute value of the minimum negative value of the spring stress curve) at different impact heights, the maximum system contact force, the maximum acceleration (after FFT treatment) and velocity of the selected rigid node of the tail beam, and the maximum acceleration and velocity of the coal-like rock sphere are extracted. Figure 29 shows the maximum system contact response.Figure 29
Maximum contact response.Figure29 shows that as the impact position changes, the maximum value of the maximum spring stress appears when the rock sphere impacts the tail beam near the support terminus of the spring. Maximum spring stress decreases first and then increases when the impact position moves away from the support terminus of the spring. And when it comes near to x = −1 m, the maximum value of the maximum spring stress appears again. When the impacted position continues to approach the rigid support terminus, the maximum spring stress decreases again and it increases slightly near the terminus. The trend of the maximum spring stress of the system, as a whole, appears decrease ⟶ increase ⟶ decrease ⟶ increase ⟶ decrease ⟶ increase (the direction is from the support terminus of the spring to the rigid support terminus, i.e., x = −1.3 m ⟶ x = −0.75 m). And the minimum value of the maximum spring stress of the system appears around x = −1.25 m. The maximum contact force of the system, as a whole, shows an increase ⟶ decrease ⟶ increase ⟶ decrease ⟶ increase trend. The maximum contact force reaches its maximum value around x = −0.95 m while its minimum value appears near x = −1.2 m. That is, the minimum value of the maximum contact force occurs at a short distance from the support terminus of the spring, and the minimum value of the maximum contact force appears close to the center of the tail beam but slightly biased toward the fixed support terminus. The maximum acceleration belonging to the selected rigid node of the tail beam shows a decrease ⟶ increase ⟶ decrease ⟶ increase trend (the oscillation changes after x > −1.2 m is considered to be decreasing all the time). The maximum value of the maximum acceleration of the selected rigid node occurs at x = −1.3 m that is closest to the support terminus of the spring. And the minimum value of the maximum acceleration occurs at x = −0.75 m which is closest to the rigid support terminus. The maximum velocity of the selected rigid node shows the trend of decreasing ⟶ increasing ⟶ decreasing ⟶ increasing ⟶ decreasing. The maximum value of the maximum velocity of the selected rigid node reaches its maximum value at x = −1.3 m which is closest to the support terminus of the spring while it reaches its minimum value at x = −1.15 m which is close to the center of the tail beam but slightly biased toward the elastic support terminus (spring). Therefore, the most obvious vibration signal of the tail beam occurs when the support terminus of the spring is impacted. The minimum and maximum values of the maximum acceleration of the rock sphere are 976307 m/s2 and 976474 m/s2, respectively. And the minimum and maximum values of the maximum velocity are 5.6572 m/s and 5.65799 m/s, respectively. In addition, the maximum changing rates of the maximum acceleration and velocity are 0.017105% and 0.013965%, respectively, which can be ignored. So, when the same rock sphere impacts the different positions of the tail beam along the central axis, its comprehensive recovery coefficient hardly changes.


## 5.3. Contact Response of the System with Different Impact Sizes

For impacted objects with the same properties, its different sizes will result in different influences to the contact stiffness, the impact energy, the equivalent contact radius, and the contact area of the system. Moreover, the compression depth of the rock sphere, the contact force, and the structural force of the system are also affected. Therefore, the contact response of the system will be different when the radius of the impacted object is various. As the varisized coal-like rock spheres impact the same position (x = −1 m) of the tail beam at the same velocity of 8.368 m/s (the same impact height), the curves of the spring stress, the system contact force, the acceleration and velocity of the rigid node of the tail beam, and the acceleration and velocity of the coal-like rock sphere are shown in Figures 30–35. And the acceleration curve of the rigid node close to the hinged holes of the tail beam (Figure 32) is processed by FFT.Figure 30
Spring stress.Figure 31
Contact force.Figure 32
Acceleration of the tail beam.Figure 33
Velocity of the tail beam.Figure 34
Acceleration of the sphere.Figure 35
Velocity of the sphere.As shown in Figures30–35, when the coal-like rock spheres with the different radius impact the same position of the tail beam at the same velocity, the curve form of the spring stress, the vibration velocity of the selected rigid node of the tail beam and the velocity and acceleration of the rock sphere is the same as the simulation time varies. As the radius of the coal-like rock sphere increases, the spring stress, the contact force between the rock sphere, and the tail beam as well as the vibration velocity and acceleration of the selected rigid node of the tail beam increase, while the velocity and acceleration of the rock sphere decrease. The reason is that the material properties remain constant, the larger the radius of the rock sphere is, the greater the Hertz contact stiffness of the system is. Furthermore, the mass and the initial impact energy of the coal-like rock sphere also increase. As a result, the contact force between the rock sphere and the tail beam increases and more energy is transferred to the tail beam when it is impacted. Therefore, the vibration velocity and acceleration of the rigid node of the tail beam increase. The energy transferred to the tail beam increases, resulting in that the ratio of the energy of the coal-like rock sphere to its initial impact energy decreases after impact contact. Furthermore, after the impact-contact process, the larger the radius of the rock sphere is, the smaller the velocity and acceleration of the rock sphere are. Similarly, the velocity and acceleration curves of the rock sphere quickly remain steady.The same treatment is also utilized to, respectively, extract the maximum values of the contact responses in Figures30–35 when the rock spheres with different radii impact the tail beams (where the maximum acceleration of the selected rigid node is equal to the maximum value of the acceleration curve after FFT treatment, and the maximum spring stress is defined as the absolute value of the minimum negative value of the spring stress curve). Then, the maximum contact response of the system at different impact heights is obtained, as shown in Figure 36. According to Figure 36, it can be seen that as the radius of the rock sphere increases, the maximum spring stress, the maximum system contact force, and the velocity of the rigid element close to the hinged hole of the tail beam increase gradually, and their growth rates increase. This shows that the larger the radius of the rock sphere is, the more obvious the vibration amplitude of the tail beam is. And the acceleration of the rigid node close to the hinged hole of the tail beam increases gradually while its growth rate decreases slightly. As the radius of the rock sphere increase, its maximum velocity and acceleration decrease and the decreasing rate gradually slows down. The impact velocities of rock spheres are 8.368 m/s, and the maximum rebound velocities of four groups of rock spheres with different radii are 5.65751 m/s, 3.96624 m/s, 3.57084 m/s, and 2.78333 m/s, respectively. And their comprehensive recovery coefficients are 0.676088671, 0.473977055, 0.426725621, and 0.332615917, respectively. When the rock sphere impacts the tail beam, the comprehensive recovery coefficient of the rock sphere decreases as its radius increases.Figure 36
Maximum contact response.


## 5.4. Contact Response of the System with Different Impact Materials

Two kinds of particles coal and gangue can be seen in the process of coal mining. And the properties of coal and gangue under different working faces are also different. The contact stiffness and the damping of the system vary with the material of the contacted object, which leads to the change of the system impact-contact response when the particles impact the tail beam. In this paper, one kind of coal and two different kinds of gangue (sandstone and marble) are considered as the research object, as shown in Table4. When the different elastic spheres (which are defined as coal elastic sphere, sandstone elastic sphere, and marble elastic sphere) with the same radius, respectively, impact the same position (x = −1 m) of the tail beam at the same velocity (the same impact height) of 8.368 m/s, the curves of the spring stress, the system contact force, the acceleration and velocity of the selected rigid node of the tail beam, and the acceleration and velocity of the coal-rock sphere are shown in Figures 37–42.Figure 37
Spring stress.Figure 38
Contact force.Figure 39
Acceleration of the tail beam.Figure 40
Velocity of the tail beam.Figure 41
Acceleration of the sphere.Figure 42
Velocity of the sphere.Figures37–42 show that when the coal elastic sphere, sandstone elastic sphere, and marble elastic sphere with the same radius impact the tail beam at the same velocity, the curve forms of the spring stress, the vibration acceleration and velocity of the selected rigid node of the tail beam and the vibration acceleration and velocity of the rock sphere are the same as the simulation time changes. When the same position of the tail beam is impacted at the same velocity, the spring stresses, and the vibration velocity of the rigid node of the tail beam produced by the impacts of sandstone and marble elastic spheres with the same radius are almost the same, the impacts of the sandstone elastic sphere are slightly smaller than those of the marble elastic sphere. When the coal-like rock sphere impacts the tail beam, the spring stress, the vibration acceleration and velocity of the selected rigid node of the tail beam, and the acceleration of the rock sphere are far less than those of the sandstone elastic sphere and the marble elastic sphere. However, the velocity law of the rock sphere is contrary (it means that when the rock sphere with the same radius impacts the same position of tail beam at the same velocity, the velocity of the coal elastic sphere is much faster than that of the sandstone elastic sphere and marble elastic sphere).Figures37–42 also show that when the coal elastic sphere impacts the tail beam, the maximum single spring stress is 33.685 MPa and the minimum single spring stress is −49.8086 MPa. And the maximum changing value of the single spring stress is 83.4936 MPa. Another, when the sandstone elastic sphere impacts the tail beam, the maximum single spring stress is 55.4426 MPa and the minimum single spring stress is −82.6176 MPa, and the maximum changing value of single spring stress is 138.0602 MPa. And when the marble elastic sphere impacts the tail beam, the maximum single spring stress is 56.7875 MPa and the minimum single spring stress is −86.0464 MPa. Its maximum changing value of the single spring stress is 142.8339 MPa. According to the data, the maximum spring stresses (which is defined as the absolute value of the minimum negative value of the spring stress curve) of the spring caused by the impacts of the sandstone and marble elastic spheres on the tail beam are 1.6587 times and 1.72754 times of the maximum spring stress caused by the impact of coal elastic sphere. And when the sandstone and marble elastic spheres impact the tail beam, their maximum changes of single spring stresses are 1.65354 times and 1.71072 times of the maximum spring stress caused by the impact of coal elastic sphere. When the gangue-like elastic sphere impacts the tail beam, the spring stress and the maximum changing value of the single spring stress are much greater than those of the impact caused by coal-like elastic sphere. At the same time, it can be known that when the coal, sandstone, and marble elastic spheres impact the tail beam, the maximum contact forces of the system are 1450.72 N, 4249.05 N, and 6341.25, respectively. And the contact forces produced by the impacts of sandstone and marble elastic spheres are 2.92892 times and 4.3711 times that of the coal elastic sphere. The maximum acceleration of the selected rigid node of the tail beams is 116.073 m/s2, 199.969 m/s2, and 264.017 m/s2, respectively. When the sandstone and marble elastic spheres impact the tail beam, their maximum acceleration of the selected rigid node is 1.72279 times and 2.27458 times those of the coal elastic sphere. Velocities of the selected rigid node are 0.000718687 m/s, 0.00122 m/s, and 0.00137 m/s, respectively. And the maximum velocities of the selected rigid node when the sandstone and marble elastic spheres impact are 1.69754 times and 1.90625 times that of the coal elastic sphere. The maximum acceleration of the rock spheres is 976369 m/s2, 1986110 m/s2, and 1140600 m/s2. In addition, the maximum rebound accelerations of the sandstone and marble elastic spheres after their impacts are 2.03418 times and 1.16821 times that of coal elastic sphere. The maximum velocities of the three rock spheres are 5.65751 m/s, 2.72292 m/s, and 3.04507 m/s, respectively. And rebound velocities of sandstone elastic sphere and marble elastic sphere after their impacts are 0.4812 times and 0.53824 times that of coal elastic sphere.According to the above data, the contact response of coal elastic sphere impacting the tail beam, such as spring stress, system contact force, the acceleration and velocity of the selected rigid node of the tail beam, and the acceleration and velocity of the rock sphere are different from those of the impact of gangue elastic sphere. In the simulation model, the spring replaces the jack of the tail beam in the solid model. On the basis of this, when the gangue elastic sphere, respectively, impacts the tail beam, the oil pressure, its variable quantity of the jack, and the vibration velocity of the tail beam after its impact are much larger than the change of the response parameters caused by the impact of coal elastic sphere, while the rebound velocity of the gangue elastic sphere particles is much smaller than that of the coal elastic sphere after the impact. Therefore, in the process of technical researches on the coal gangue identification for longwall top coal caving or on the coal gangue identification of particles impacting the tail beam of the hydraulic support, the oil pressure of the jack of the tail beam after impact can be extracted by the pressure sensor and its changing value can be monitored. The vibration response of the tail beam after impact can be extracted by the velocity or the acceleration sensor, and the rebound velocity of the rock sphere can be pictured by using instruments such as high-speed camera. It is feasible to carry out the gangue identification in accordance with the differences of the measured parameters (note: for a qualitative research, both the coal and gangue particles are equivalent to elastomers in the simulation. Actually, they are brittle objects and coal is easier to break than gangue particles. In actual impact experiment, when the coal and gangue particles impact the tail beam, the differences of the oil pressure and its variable quantity of the jack of tail beam, as well as the vibration velocity of the tail beam after impact will be greater. Based on these parameters, the realizability to perform the coal gangue identification will be higher. However, because of the fact that coal is easier to break than gangue particles, the rebound velocity of coal may be smaller than that of gangue in the actual experiment.)


## 6. Conclusion

In the mining of longwall top coal caving, the impact-contact behavior between the coal gangue particles and the tail beam of the hydraulic support often occurs. In order to accurately grasp the dynamic response system of the coal gangue impacting the tail beam, a simplified device model and kinematics model of the tail beam were established in this paper. Furthermore, by combining the equivalent stiffness theory of the jack of the tail beam, the contact model of L-N nonlinear spring damping, and the energy conservation law, the system contact-structure dynamic model of the particles impacting the massless tail beam of hydraulic support was established. On this basis, in order to accurately study the dynamic contact response of the system, the method of finite element simulation was used to carry out the four different types of finite element dynamic simulations: impacts of coal-like elastic particles on the massless tail beam from different heights, coal-like elastic particles on the different positions of central axis of massless tail beam, coal-like elastic particles with different radii on the massless tail beam, and coal gangue-like elastic particles with different materials on the massless tail beam. In a word, the conclusions are as follows:(1)
According to the contact-structure dynamics model of the particles impacting the tail beam of the hydraulic support established in this paper, the system contact response is related to the parameters such as the mass of the impacted object, the properties of the contacted object, the impact height, the impact position, and the initial position of the tail beam when the rock sphere impacts the tail beam.(2)
As the impact height increases, the impact-contact velocity of the same rock sphere (i.e., the initial impact energy) increases. When the same position of the tail beam is impacted, the spring stress of the system, the contact force between the rock sphere and the tail beam, the vibration velocity and acceleration of the selected rigid node of the tail beam, and the velocity and acceleration of the rock sphere increase. And the amplitude of the above contact responses increases accordingly.(3)
The support form of the tail beam consists of the rigid support and the flexible support at the different termini, which can be counted as the composite support. It causes that the structural stress of the system impact response changes when the coal-like rock sphere with the same radius impacts the different positions of the tail beam at the same velocity. And the contact response of the system does not change monotonously when the impact position changes. Maximum rebound velocity and acceleration of the rock sphere are approximately the same after its impact on the different positions of the tail beam, and its residual energy after impact is almost unchanged. When the impact position is located near the rigid support terminus, the maximum value of the maximum spring stress occurs. And when the impact position is close to the middle of the tail beam but slightly biased toward the flexible support terminus, the maximum contact force occurs. Overall observation shows that the vibration amplitude of the tail beam is smaller when the impact position moves to the rigid support terminus. Besides, when the flexible support terminus of the tail beam is impacted, the maximum value of the maximum vibration velocity of the tail beam appears. And when the impact position is close to the middle position of the tail beam (x = −1.05 ⟶ −0.95), the curve trends of some system contact responses such as the maximum contact force, the maximum spring stress, the vibration velocity, and the acceleration of the tail beam change.(4)
For the elastic sphere with the same material, as the radius of the sphere increases but the material properties remain constant, the Hertz contact stiffness of the system as well as the mass of the sphere increases. In addition, the increase also goes to the initial impact energy of the sphere with the same velocity. When the same position of the tail beam is impacted, the contact force between the rock sphere and the tail beam increases, as well as the energy transferred to the tail beam. Therefore, the vibration amplitude of the tail beam after impact grows while the ratio of the energy of the sphere after impact to its initial impact energy reduces. And the comprehensive recovery coefficient (which is defined as the ratio of the maximum rebound velocity after impact to the initial impact velocity) decreases as the radius of sphere increases.(5)
When the coal and gangue elastic spheres (with the same radius and different materials) impact the same position of the tail beam at the same velocity, the spring stress produced by the impact of coal-rock sphere on the tail beam, the vibration acceleration and velocity of the rigid node of the tail beam, and the acceleration of the rock sphere are much smaller than those of sandstone elastic sphere and the marble elastic sphere. The simulations only take the elastic properties of the sphere into consideration but fail to consider the brittleness of the rock sphere, so the rebound velocity of the coal elastic sphere is much faster than that of sandstone elastic sphere and marble elastic sphere according to the simulation results. In reality, both coal and gangue are brittle objects. In the actual experiment of the gangue impacting the tail beam, they will be damaged and broken among which the coal is easier to break than gangue. In the actual working condition of the coal and gangue impacting the tail beam, the differences of the oil pressure and its variable quantity of the jack of tail beam, the vibration velocity differences of the tail beam after impact, and the rebound velocity differences of the particles after impact will be greater. Based on the differences of these contact responses, it will be more realizable for coal gangue identification.In this paper, a four-in-one contact-structure impact dynamic model composed by the rock sphere, the tail beam, the jack of the tail beam, and the supporting frame is established by combining the solid contact theory with the structural mechanics of the tail beam. This research provides a theoretical basis for the impact between the particles and the complex support structures and supports a simulation method in finite element software for the contact analysis of the system which contains the hydraulic cylinder. It also proves the feasibility of identifying coal gangue based on the differences of the system contact response after the impact by coal gangue, as well as offers a research basis for the realization of identification technology of the coal gangue in the longwall top coal caving. In the future research, the brittleness of coal gangue particles will be considered. And under this consideration, the system contact response and response differences of the coal gangue impact on the tail beam will be studied.



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*Source: 6030542-2019-10-27.xml*