J. Cent. South Univ. (2021) 28: 2120-2130

DOI: https://doi.org/10.1007/s11771-021-4757-z

Nonlinear multi body dynamic modeling and vibration analysis of a double drum coal shearer

ZHANG Xiao-li(张晓丽)^{1, 2}, YAO Guo(姚国)¹, ZHANG Yi-min(张义民)³

1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China;

2. School of Mathematics, Changchun Normal University, Changchun 130032, China;

3. Equipment Reliability Institute, Shenyang University of Chemical Technology, Shenyang 110142, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract: The double drum coal shearer is widely applied for the underground coal exploration in the mining industry. The vibration and noise control are significant factors for the stability design of the double drum coal shearer. In this paper, the vibration properties of a double drum coal shearer are firstly investigated. The horizontal, transverse and torsional vibrations of the motor body and the angle displacements of the rockers are taken into account. The walking units and the hydraulic units are modeled by the stiffness-damping systems. The nonlinear equation of motion of the double drum coal shearer is established by applying the Lagrange’s equation. The nonlinear vibration response of the system is calculated by using the Runge Kutta numerical method. The effects of the shearing loads, the equivalent damping and stiffness of the walking units, the inclination angels of the rockers and the equivalent damping and stiffness of the hydraulic units on the vibration properties of the system are discussed.

Key words: coal shearer; vibration; multi body dynamics; transient response

Cite this article as: ZHANG Xiao-li, YAO Guo, ZHANG Yi-min. Nonlinear multi body dynamic modeling and vibration analysis of a double drum coal shearer [J]. Journal of Central South University, 2021, 28(7): 2120-2130. DOI: https://doi.org/10.1007/s11771-021-4757-z.

1 Introduction

The coal mining industry is closely related to the national economy and the people’s livelihood for most countries all over the world. The coal shearer is normally composed of the travelling unit, the shearing unit and the engine body [1], which is designed for the underground coal exploration since 1950s [2, 3]. For a coal shearer system, the shearing loads generated from the coal cutting transmit from the rockers to the engine body of the coal shearer and to the travelling unit. The high-circle vibration of the coal shear will do harm to the safety and reliability of the system. So, the dynamics of the coal shearer as well as its components have attracted many researchers’attention.

GAO et al [4] researched the effect of dip angle in mining direction on drum loading performance by using the discrete element method. The improvement suggestions were proposed for the design of the drum coal loading rate. YANG et al [5] studied the effects of the cutting speed on the dynamic characteristics of the long wall shearer by applying the AMESim software and found that the system with variable cutting speed exhibit more stable dynamic characteristics. ZHANG et al [6] investigated the vibration properties of a three-drum shearer for deep mining. The shearer was designed and analyzed by the FEM. The vibration characteristics of the front drum was computed and some design suggestions were given for the three-drum shearer design. ZHANG et al [7, 8] studied the dynamic properties of the traction unit gear system of a long wall coal shear. The vibration displacements were numerically obtained and compared with the experiment results. CHEN et al [9] researched the vibration properties of the gear transmission system of a shearer. The reliability of the system under stochastic excitation was discussed. WANG et al [10] studied the vibration characteristics of a drum shearer gearbox subjected to oblique cutting, and exhibited the relationship between the cutting depth and the vibration amplitude of the system.

JIANG et al [11] studied the torsional vibration characteristics of the transmission system with internal parameter and external cutting load excitation changes. The vibration jump phenomenon of the nonlinear electromechanical coupling system was analyzed. SHENG et al [12] studied the bifurcation and chaotic motion of a shear. Their results showed that bifurcation and chaotic motion can be controlled effectively when reasonable parameters were concerned. YANG et al [13] established the dynamic model of the cutting unit of the drum shearer. The dynamic properties of the system were analyzed by simulation. GAO et al [14] established the model of shearer drum with several arranged picks. The influence of the rock distribution on cutting force was investigated. CHEN et al [15] established the rigid-flexible coupling model of a shearer and found the force on the right side of the plane support plate was smaller than that on the left side. ZHOU et al [16] studied the dynamic reliability of planetary gear system in shearer mechanisms and concluded that the sun gear affected the dynamic performance more than other gears in the planetary gearbox. GAO et al [17] set up the dynamic analysis model of the elastic shaft of the cutting part and the critical speed of the elastic shaft system was analyzed.

WANG et al [18] proposed a closing path optimal estimation model of the longwall mining and improved the positioning accuracy. GE et al [19] proposed an active torque compensation control method to control the effects of a load-induced gear dynamic load of drum shearer. JIA et al [20] proposed a 3D gear model to accurately calculate the meshing force of the gear system of shearer and concluded that it was better applied for the wide-faced gear. LI et al [21] set up the rigid-flexible coupling multibody model of the cutting unit of shearer. The vibration properties of the pulling, vertical, horizontal and torsional directions were analyzed. GAO et al [22] established the dynamic model of the rocker arm and studied the free vibration of the system. LIU et al [23] carried out the experiment test of the shearer walking unit. The test bed was built and the vibration characteristics of the walking unit was tested. ZHAO et al [24] applied Labview for data acquisition and signal analysis to collect and analyze the shearer body’s vibration data. ZHAO et al [25] investigated the vibration of shearer and found that cutting pick was the disturbance source of the noise. TONG et al [26] studied the nonlinear dynamic contact properties of the gear stress in LS-DYNA. CAMARGO et al [27] developed the dynamic model of a single cutting drum and analyzed the noise distribution of the system.

From the literatures reviewed above it can be seen that the vibration properties of the drum shearer were studied mostly for the single parts of the machine. The vibration properties of the whole machine of the shearer are scarcely seen in the existing investigation. In fact, by analyzing the dynamic properties of the whole machine of the coal shear, the energy transformation path of the system can be clearly understood and the weaklink of the machine can be identified. The analyzing results of the dynamics of the whole machine of the coal shear can give guidance for the harmonic optimal design for the two drum coal shear. Meanwhile, the structure of the coal shearer is a complex multi body system which may contain some novelty dynamic phenomenon. This motivated us to carry out the current study. In this paper, the nonlinear forced vibration model of a double drum coal shearer is firstly set up by using the Lagrange’s equation. The structural vibration properties of the coal shearer can be obtained by solving the nonlinear ordinary differential equations. The solving method and discussion scope are common with current studies on other structural dynamic problems [28, 29]. In this study, the vertical, horizontal and torsional displacements of the system are considered. The displacement responses of the shear arm and the machine body are displayed and the variation of the dynamic properties of the response with respect to the design parameter of the coal shear are discussed. Compared with the existing investigations, this paper can provide design suggestions for the coal shearer from the perspective of the whole system dynamic and stability.

2 Dynamic modeling of double drum shearer

The schematic diagram of the double drum shearer is shown in Figure 1. The double drum coal shearer is composed of cutting unit, machine body and traveling unit. The rockers manipulating the drum are supported and controlled by the hydraulic cylinders. The external excitation originated from the cutting process of the cutting picks transfers from the rockers to the engine body of the shearer.

Figure 1 Schematic diagram of a double drum coal shearer

In order to establish the whole vibration model of the double drum coal shearer, the physical model of the shearer is equivalent to a kinetic model as shown in Figure 2. In the kinetic model, the engine body is equivalent to a rigid mass with length L and height H. The horizontal, vertical and torsional displacements of the rigid body are taken into account and they are expressed by u, w and θ. The mass density of the rigid body is m, kg/m³, from which the mass and the rotational inertia can be calculated. The walking units of the double drum coal shearer are modeled as the spring-damping systems with elastic and damping coefficients k₁, k₂, c₁ and c₂. The displacement between the two walking units is 2R. The rockers are fixed on the engine body and controlled by the hydraulic cylinders. The mass of the rockers and the drums are concentrated on the mass points m₁ and m₂. The length of the rockers are L₁ and L₂. The distances from the fixing point to the supporting point of the hydraulic cylinders are R₁ and R₂. The mining heights of the cutting units are depended on the angles between the rocker arms and the horizontal line, which are expressed as α₁ and α₂. The deflections of the rocker arms are expressed by φ₁ and φ₂. The length of the hydraulic cylinders are b₁ and b₂. The equivalent stiffness and damping coefficients are k₃, k₄, c₃ and c₄.

The movement of the structure can be described by the generalized coordinate vector q=[u, w, θ, φ₁, φ₂]^T. The vertical displacement of the walking units of the coal shearer can be expressed by:

                      (1)

where x₁ and x₂ are the vertical displacements of the left and the right walking unit.

The absolute deformation of the walking units considering the effects of the axial and the vertical deformations can be obtained by using the Pythagorean theorem. Taking the left foot of the coal shearer shown in Figure 3, the deformation of the walking unit can be calculated by the length after the

deformation minus the length before deformation. So the absolute deformation of the springs of the walking units are expressed as:

     (2)

where s₁ and s₂ are the deformations of the equivalent springs of the left and right walking units.

The deformation of the equivalent springs and damping system of the hydraulic cylinder are expressed as:

       (3)

where s₃ and s₄ are the deformation of the equivalent spring deformation of the hydraulic cylinders.

Figure 2 Kinetic model of double drum coal shearer

Figure 3 Absolute deformation of left walking unit of coal shearer

The Lagrange’s equation is adopted to establish the equation of motion of the system. In such case, the potential energy U, the kinetic energy T and generalized non-conservative forces should be calculated first.

The potential energy of the system U can be written as:

              (4)

The mass of the rigid body can be expressed as

                          (5)

where M is the mass of the rigid body. The rotational inertia of the double drum coal shearer can be written as:

           (6)

where J is the rotational inertia. The kinetic energy of the system energy can be expressed by

                     (7)

where T is the kinetic energy and ‘·’ is derivative with respect to time.

In this study, the cutting loads of the drum are expressed by the summation of a static load and a sinusoidal load with different amplitude, frequency and phase angles. This assumption is from the experiment results in China Coal Zhangjiakou Coal Mining Machinery Company of Limited Liability. In the experiment, a simulative coal wall was built for a height of 3 m. The model of the coal shearer was MG500/1180 and the velocity of cutting was 4 m/min. Figure 4 shows the experiment system of the fully mechanized mining face and Figure 5 shows the tested load of the drum and its frequency spectrum. From Figure 4 it can be seen that the cutting force is composed of a static force and a series of alternating loads.

Figure 4 Experiment system of fully mechanized mining face

According to the above experimental result, the loads acting on the drum are assumed to be composed of the vertical load and the horizontal load, and they are expressed as:

                    (8)

where f₁ and f₂ are horizontal and vertical loads acting on the left drum, respectively; and f₃ and f₄ are horizontal and vertical loads acting on the right drum. By decomposing these excitations and the damping forces to the generalized coordinates, one can obtain the generalized excitations of the system.

 (9)

Figure 5 Experimental shear force of drum in experiment (a) and corresponding frequency spectrum (b)

where W_u is the generalized force associated with the horizontal direction of the rigid body. The generalized force along the w direction is

 (10)

The generalized force along the θ direction is

                   (11)

The generalized forces along the φ₁ and φ₂ directions are expressed as

     (12)

   (13)

Substituting the potential energy, the kinetic energy and the generalized forces to the Lagrange’s equation

      (14)

where q_j is the generalized coordinates standing for u, w, θ, φ₁ and φ₂ in this paper; F_j is the generalized excitation of the system which are expressed by W_u,W_w, W_θ, W_φ1 and W_φ2 in this paper. By using the Lagrange’s equation, one can obtain the nonlinear equation of motion of the present system.

                              (15)

               (16)

                 (17)

         (18)

        (19)

Equations (15)-(19) are the nonlinear dynamic models of the double drum coal shearer. The dynamic model of the shearer is nonlinear ordinary differential equation with 5 degrees of freedom. In the present study, in order to obtain the detailed vibration characteristics of the system, the nonlinear ordinary differential equation is numerically solved by using the fourth order Rung-Kutta method.

In this study, the equation of motion of the coal shearer is numerical solved by using the fourth order Runge-Kutta method, which is a robust and effective numerical method for solving the nonlinear ordinary differential equation of motion. By transforming the equation of motion of the shearer into the state equation and importing to the Matlab software, one can obtain the displacement responses of the coal shearer.

3 Numerical simulations and discussions

A double drum coal shearer shown in Figures 1 and 2 is considered. The structural and the equivalent mass, stiffness and damping coefficients are H=0.7 m, L=4 m, L₁=1.5 m, L₂=1.5 m, R₁=0.5 m, R₂=0.5 m, h=0.05 m, R=1.5 m, α₁=60°, α₂=30°, b₁=0.3 m, b₂=0.3 m, k₁=5000000 N/m, k₂=5×10⁶ N/m, k₃=2×10⁶ N/m, k₄=2×10⁶ N/m, c₁=10000 N·s/m, c₂=10000 N·s/m, c₃=1000 N·s/m, c₄=1000 N·s/m, m₁=200 kg, m₂=200 kg and m=1000 kg/m².

In the numerical simulation, the cutting excitations are modeled by static load with addition to a harmonic excitation. In the engineering practice, the static loads are largely determined by the cutting depth, and the amplitude and frequency of the harmonic excitations are determined by the rotating speed of drums. In the simulation, a phase angle is introduced to distinguish the vertical and horizontal harmonic excitations. As the purpose of the present study is to investigate the overall vibration characteristics of the coal shear, the detailed relationships between the cutting depth and the static load, and the rotating speed and the harmonic excitation are not especially discussed. The equivalent loads are adopted just by practice experiences.

It also should be clarified that the equivalent spring and damping coefficients are from the engineering practice, and they are variables in some discussions. The detailed relationships between the physic properties of the hydraulic cylinder, the walking units and the spring-damping system are not within the scope of the present study.

Figure 6 shows the displace response of the structure under the cutting loads. In the simulation, the excitation parameters are chosen as F₁=F₂=F₃=F₄=500 N, F₁₀=F₂₀=F₃₀=F₄₀=800 N, ω₁=ω₂=ω₃=ω₄=6 rad/s, β₁=0°, β₂=90°, β₃=0°, β₄=90°. The time for simulation is 30 s. It can be seen from Figure 6 that the responses of the engine body are composed of transient responses and steady state responses. The time for transient response of the horizontal, vertical and torsional directions is about 15 s. It also can be seen in Figure 6 that the transient response originated from the effects of the static load and the harmonic excitation disappears after a period of time for the damping effect of the system. Then only the steady state response of the system remains. The amplitude of the horizontal transient vibration is 0.05 m and the center for the horizontal transient vibration is at u=0. The steady state response of the horizontal displacement of the engine body appears when t=15 s. The center of the steady state response is at u=-0.04 m.

Figure 6 Displacement responses of generalized coordinates of double drum coal shearer

Figure 7 shows the local amplification of the steady state response of the horizontal displacement. It can be seen that the steady state response of u is largely dependent on the harmonic response with amplitude of 0.01 m.

The steady state responses of w and θ have similar properties as those of u. The center of the steady state response of w is at w=-0.05 m and the amplitude of the steady state response of w is 0.005 m. The center of the torsional displacement of the engine body is at θ=-0.021 rad, and the amplitude of the torsional displacement θ is 0.0023 rad.

Figure 7 Local amplification of transient response of horizontal displacement of engine body

The responses of the rockers have shorter period of transient responses than those of the engine body. After 6-8 s of the transient vibration, the responses of the left and the right rocker become steady state responses. The steady state amplitude of the left rocker and the right rocker is 0.0025 rad.

In order to examine the effects of the rocker inclination angle on the dynamic response of the system, the inclination angle of the right rocker is taken as variable and the displacements of the horizontal direction for different inclination angle are compared. Figure 8 shows the comparison of the displacements of u direction when α₂=30° and α₂=45°. Except for the inclination angle of the right rock, the other parameters are the same as those in Figure 6. It can be seen in Figure 8 that the period for transient vibration when α₂=45° is 12 s, which is smaller than that when α₂=30°. Figures 9 and 10 show the comparisons of the displacement and angle response of the system for α₂=30° and α₂=45°. It can be seen in Figures 9 and 10 that the dynamic properties of w and θ are similar as that of u.

Figure 8 Displacement responses of u for different inclination angle of right rockers

Figure 9 Displacement responses of w for different inclination angle of right rockers

Figure 10 Angle responses of θ for different inclination angle of right rockers

The effects of the equivalent stiffness of the hydraulic cylinders on the dynamic properties of the system are discussed as follows. The parameters remain the same as those in Figure 6 except for k₃ and k₄. Figure 11 compares the displacement responses of u for different k₂ and k₃.

Figure 11 displays the effect of the equivalent stiffness of the hydraulic cylinders on the dynamic response of u. It can be concluded that when k₂=k₃=2.5×10⁶ N/m, the amplitude of the transient vibration is lower than that when k₂=k₃=2×10⁶ N/m. However, the time period for transient vibration when k₂=k₃=2.5×10⁶ N/m is 16 s, which is longer than that when k₂=k₃=2×10⁶ N/m. Figures 12 and 13 display the effect of the stiffness coefficient of the hydraulic cylinder on the responses of the vertical vibration and torsional vibration of the engine body. It can be seen that the vertical vibration and the torsional vibration exhibit similar characteristics as that of the horizontal vibration.

Figure 11 Displacement responses of u for different equivalent stiffness of hydraulic cylinders

Figure 12 Displacement responses of w for different equivalent stiffness of hydraulic cylinders

Figure 13 Angle responses of θ for different equivalent stiffness of hydraulic cylinders

Figure 14 exhibits the effect of the equivalent damping coefficient of the hydraulic cylinders on the horizontal vibration properties of the engine body. In Figure 14, the solid line exhibits the displacement when c₂=c₃=950 N·s/m and the dashed line describes the displacement of u when c₂=c₃=1000 N·s/m. It can be seen in Figure 14 that the time period for transient vibration of u is 60 s when c₂=c₃=950 N·s/m, which is much longer than that when c₂=c₃=1000 N·s/m. It also can be seen from Figure 14 that the two curves are similar when t<10 s. This is because the two systems for different damping coefficients have the same initial conditions and external loadings. The systems responses are similar at the initial moment. After a period, the transient response reduces for the existence of the damping parts. The discrepancies of the responses accumulate with time increasing and then go apart. The displacements vary significantly when t>10 s, which means that the vibration system of the double drum coal shearer is a strong nonlinear dynamic system.

Figure 14 Displacement responses of u for different equivalent damping coefficients of hydraulic cylinders

Figures 15 and 16 show the displacement and angle responses of the vertical and torsional directions of the system. It can be seen from Figures 15 and 16 that the vibration properties of the these directions have similar properties as that of Figure 14.

4 Conclusions

This paper considers the vibrations of a double drum coal shearer. The displacements of the horizontal, vertical and torsional directions and the rocker angles are taken into account. The nonlinear dynamic model of the system with 5 degrees of freedom is established by using the Lagrange’s equation. The dynamic model is solved by using the numerical simulation and the following conclusions are drawn.

Figure 15 Displacement responses of w for different equivalent damping coefficients of hydraulic cylinders

Figure 16 Vibration responses of θ for different equivalent damping coefficients of hydraulic cylinders

1) The response of the system is composed of the transient response period and the steady state response period. The length of the period of time is associated with the excitation parameters and the structural parameters.

2) The period for transient vibration when α₂ = 45° is smaller than that when α₂ = 30°, which means that the mining height has significant effect on the stability of the system.

3) With the equivalent stiffness of the hydraulic cylinder increasing, the length of the time period for transient vibration of the system is prolonged. But the amplitude of the transient vibration decreases with equivalent stiffness coefficient increasing.

4) With the equivalent damping coefficient of the hydraulic cylinder increasing, the time period of the transient vibration of the system decreases.

Contributors

YAO Guo provided the research background of the paper and edited the draft of manuscript. ZHANG Xiao-li conducted the literature review and wrote the first draft of the manuscript. ZHANG Yi-min provided the experiment conditions.

Conflict of interest

ZHANG Xiao-li, YAO Guo, and ZHANG Yi-min declare that they have no conflict of interest.

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(Edited by FANG Jing-hua)

中文导读

双滚筒采煤机的非线性多体动力学建模及振动分析

摘要：双滚筒采煤机广泛应用于煤矿工业中的地下煤炭开采。采煤机的振动和噪声控制是对采煤机进行稳定性设计的重要因素之一。本文对双滚筒采煤机的多体动力学振动特性进行分析。考虑采煤机机身横向、纵向和扭转位移，以及采煤机摇臂的转角，将采煤机行走部以及液压缸模拟为弹簧-阻尼系统，采用Lagrange法建立了双滚筒采煤机的非线性多体动力学模型，并用龙格库塔法进行数值求解。讨论了截割载荷、行走部的等效弹性-阻尼系数、摇臂转角以及液压缸的等效弹性-阻尼系数对系统振动行为的影响。通过仿真结果提出了对于双滚筒采煤机稳定性的若干设计建议。本文可为煤炭开采机械的设计提供依据。

关键词：采煤机；振动；多体动力学；瞬态响应

Foundation item: Projects(51975511, U1708254) supported by the National Natural Science Foundation of China; Project(N2003023) supported by the Fundamental Research Funds for the Central Universities of China; Project(2019YFB2004400) supported by the National Key Research and Development Program of China; Project(2020-MS-092) supported by the Natural Science Foundation of Liaoning Province, China

Received date: 2020-06-10; Accepted date: 2020-12-10

Corresponding author: YAO Guo, PhD, Associate Professor; Tel: +86-13354235685; E-mail: yaoguo@me.neu.edu.cn; ORCID: https://orcid.org/0000-0001-6017-7674; ZHANG Yi-min, PhD, Professor; Tel: +86-24-83689169; E-mail: zhangyimin@syuct.edu.cn; ORCID: https://orcid.org/0000-0001-9356-8663