Strojniški vestnik - Journal of Mechanical Engineering 67(2021)1-2, 53-66 © 2021 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2020.6998 Original Scientific Paper Received for review: 2020-10-19 Received revised form: 2020-12-18 Accepted for publication: 2021-01-13 Dynamic Analysis of Hydraulic Support with Single Clearance Qingliang Zeng12 - Yangyang Li1 - Yang Yang1* 1 Shandong University of Science and Technology, College of Mechanical & Electronic Engineering, China 2 Shandong Normal University, China Hydraulic support is a complex parallel mechanism composed of multiple kinematic pairs. In the work of hydraulic support, clearances between kinematic pairs are inevitable, which lead to the deterioration of the support's working performance, and may even affect support's normal work in serious cases. To study the influence of clearances on hydraulic support and simultaneously avoid the mutual interference between different clearances, the normal and tangential force models of kinematic pairs with clearance are established, based on the Lankarani-Nikravesh contact force model and the Coulomb friction model. Furthermore, the dynamic model of the hydraulic support with single clearance is established by adding a clearance between the rear link and base and, according to the dynamic model of the hydraulic support, the research about the influence of clearance on the movement and force of hydraulic support is carried out, which proves the need to study the clearance of hydraulic support. Moreover, the influence of clearance, clearance size, and different oil inlet drive modes of front and rear columns on the stability of hydraulic support are analysed, and then the change of hydraulic support posture caused by clearance and clearance size is considered. The results show that clearance causes the movement and force of the rear link to fluctuate, which affects the stability of the hydraulic support. The clearance size affects the fluctuation degree of the movement and force, which is an important factor in the instability of hydraulic support. Also, the hydraulic support posture is changed by the clearance and clearance size. Different column oil inlet drive modes have different impacts on the clearance and have different effects on the stability of hydraulic support. Keywords: dynamic analysis, kinematic pair clearance, clearance size, hydraulic support posture Highlights • Taking the hydraulic support as the research subject, the dynamic analysis of the hydraulic support with clearance between the rear link and the base is carried out based on the Lankarani-Nikravesh model. • The dynamic performance of hydraulic support with clearance and without clearance has been compared, proving the necessity of research on hydraulic support clearance. • The dynamic performance of hydraulic support with clearance under different clearance size and oil inlet drive mode has been compared, and the influence of clearance size and oil inlet drive mode on hydraulic support has been analysed. • The influence of clearance and clearance size on the posture of hydraulic support has been studied, and the influence of posture on the working performance of hydraulic support has been analysed. 0 INTRODUCTION As important supporting equipment in coal mining, hydraulic support has always been a research hotspot in the coal industry worldwide. With the introduction of the concept of intelligent mining, hydraulic support has begen to develop in the direction of intelligent posture monitoring [1], intelligent working face perception [2] and [3], intelligent control [4] and [5], etc. Moreover, support adaptability analysis [6] and load-bearing strength analysis [7] based on experimental verification and virtual simulation also tend to the working conditions, and the analysis and research of the new hydraulic support have also made progress [8]. As an inevitable component in the production process of hydraulic support, the impact of clearance on the working performance of hydraulic support is difficult to eliminate. Therefore, it is essential to study the influence of clearance on hydraulic support for improving the working performance and the stability of the hydraulic support. However, the research on the dynamic analysis of the hydraulic support with clearance is scarce at present, and it is urgent that it be carried out. The key point of the dynamic analysis of mechanism with clearance is the model of the establishing contact force. At present, according to the different descriptions of the state between shaft and bearing, three kinds of contact force models are proposed: the continuous contact model, the two-state model, and the three-state model. Based on the contact force models, many research studies have been carried out by domestic and foreign scholars. Flores [9] and Flores et al. [10] studied the crank and analysed the influence of revolute pair with clearance on crank slider through numerical analysis and experiment. Li et al. [11] established a dynamic model of the spring operating mechanism based on the continuous contact model and obtained the influence of clearance on the movement state through a numerical solution. Pereira et al. [12] studied the influence of spherical and cylindrical contact force models on the crank slider *Corr. Author's Address: Shandong University of Science and Technology, College of Mechanical & Electronic Engineering, Qingdao, China, yang.yang@sdust.edu.cn 53 Strojniski vestnik - Journal of Mechanical Engineering 67(2021)1-2, 53-66 revolute pairs, which further improved the validity of the modelling of the revolute pairs. Li et al. [13] established a dynamic model of a connecting rod with clearance between pairs based on the two-state model and analysed the influence of different factors on the movement and force of the mechanism. Li et al. [14] established the kinematics and dynamics model of the 5-prismatic-spherical-spherical/universal-prismatic-universal (5-PSS/UPU) parallel mechanism with spherical pair clearance through the two-state model, and analysed the impact of clearance size on the dynamic performance of the system. Erkaya [15[ and [16] used the nonlinear spring damping model to conduct kinematics and dynamics analysis on the welding robot with clearance, and studied the impact of clearance on the movement error, vibration, and force of the welding robot. Bai et al. [17] established a new contact force model of planar linkage mechanism, and proposed a kinematics and dynamics optimization scheme of linkage structure to reduce the vibration peak caused by clearance. Wang et al. [18] and [19] proposed an improved normal contact force model and modified Coulomb friction model and took the crank slider and four-bar linkage mechanism as research objects to analyse the impact of clearance on the dynamic characteristics of the mechanism. Matekar and Fulambarkar [20] established a crank slider simulation model without clearance or adjustable clearance by Adams, analysed the impact of clearance on the slider displacement, and verified the simulation results through experimental devices. Tajaril et al. [21] established the mathematical model of the mechanical system by adding spherical clearance to a hexapod robot, obtained the impact of clearance on the position accuracy of the robot and verified the results through experiments. Xu et al. [22] studied the influence of clearance on the positional accuracy of the SCARA robot by establishing joint clearance model and experimental verification and analysed the determinants of clearance error. Gu et al. [23] carried out a dynamic analysis on the space manipulator with clearance based on Lankarani-Nikravesh contact force model and obtained the influence of clearance on the output results of the manipulator. Huang et al. [24] considered both the clearance and flexibility of the mechanism, and established the dynamic model of the 3-revolute-revolute-revolute (3-RRR) parallel mechanism by Adams to analyse the influence of different clearance on the displacement and velocity of the mechanism. Hou et al. [25] established a flexible 3-revolute-spherical-revolute (3-RSR) parallel mechanism simulation model based on Adams, and discussed the dynamic characteristics of the system under different clearance types. Zhang and Zhang [26] proposed that a redundant drive 4-RRR mechanism could effectively control the joint clearance through kinematics and dynamics analysis of the 3-degree of freedom (3-DOF) redundant drive 4-RRR mechanism. Erkaya [27] established a dynamic simulation model and experimental device of the spatial slider-crank mechanism system with clearance, and obtained the influence of clearance size and driving speed on the vibration response of the system. Based on an experimental device, Erkaya Fig. 1. Structure of four columns hydraulic support for top coal caving; a) front columns, b) rear column, c) top beam, d) front linkage, e) shield beam, f) tail beam jack, g) tail beam, h) plug plate jack, i) plug plate, j) rear linkage, k) shaft, and l) base 54 Zeng Q.L. - Li, Y.Y. - Yang, Y. Strojniski vestnik - Journal of Mechanical Engineering 67(2021)1-2, 53-66 [28] analysed the influence of clearance on the current fluctuation and power consumption of the system actuator, and designed a neural predictor to predict and evaluate the current. Chen et al. [29] established the planar model of hydraulic rock-breaker with multi clearance based on the Lankarani-Nikravesh model and Lagrange multiplier method, analysed the dynamic response of hydraulic rock-breaker and studied the influence of clearance size, friction and other factors on the dynamic characteristics of hydraulic rock-breaker. Tian et al. [30] compared and analysed the planar and spatial modelling methods of mechanical system with clearances, introduced the modelling methods of different types of dry and lubrication clearances, and summarized the previous clearance analysis and experiments. Edler et al. [31] developed a one-dimensional (1-D) simulation model of the progressive flow control and used in hydrostatic bearing to obtain a constant bearing clearance. Previous studies mainly focus on the analysis of clearance, and the research object is mostly simple linkage mechanism, and research on construction machinery is less common. This paper will take the hydraulic support in coal mining as the research object, combined with the actual working situation, analyse the influence of the clearance on the working performance of the hydraulic support, and provide a new direction for further improving the working performance of the hydraulic support. Hydraulic support is a complex parallel mechanism containing spherical pair, moving pair and revolute pair. In practical work, the normal operation of hydraulic support is restricted due to the unavoidable clearances. Fig. 1 is the structure diagram of the hydraulic support. It can be seen that the positions of support kinematic pair are different, as are the clearance forms of different kinematic pairs. Moreover, the kinematic pairs with clearance affect each other. If all the clearances of hydraulic support are considered at the same time, it is difficult to describe the influence of each clearance accurately. Therefore, the research of hydraulic support with clearance should start from the single clearance, and the selection of the clearance position is also very important for the research. As a parallel mechanism driven by oil cylinder, the hydraulic support controls the movement of the top beam through the front columns and rear columns, and controls the movement of the tail beam and the plug plate through the tail beam jack and the plug plate jack. The four-bar linkage mechanism composed of front and rear linkages, shield beam and base, which simultaneously affect the position change of top beam, tail beam and plug plate, which is the key to control the movement of the hydraulic support. Moreover, the rear linkage is the only single connected part containing the clearance of revolute pair in the hydraulic support, and it is an important part for the hydraulic support to bear the load. Therefore, the clearance between the rear linkage and the base is the best position to study the influence of single clearance on the hydraulic support. By establishing the dynamic model of hydraulic support with clearance between the rear linkage and base, this paper will discuss the influence of single clearance on the movement and force of hydraulic support, so as to provide theoretical and research basis for the subsequent research work of hydraulic support with clearance. 1 MODELLING OF HYDRAULIC SUPPORT WITH SINGLE CLEARANCE 1.1 The State Model at Single Clearance The key point of analysing the mechanism with clearance is the establishment of a contact force model. In order to establish the contact force model between the rear linkage and base, the contact state of bearing and shaft should be described first. Fig. 2 is a simplified model of the clearance between the rear linkage and the base. In the ideal situation, the centre of shaft and bearing are in the same position; however, due to the existence of clearance, there is an eccentric distance between the shaft and the bearing in practical work. In the figure, Oj and O2 are the centre of the shaft and the bearing, respectively, and their radii are R1 and R2, so the clearance radius of shaft and bearing can be expressed as follows: c = R2 -R,. (1) Under the fixed coordinate system O-XY, the position vectors of O1 and O2 are rj and r2 respectively, then the offset vector between shaft and bearing can be expressed as: e = r - r,. (2) In addition, the offset distance between the shaft and bearing can be expressed as the norm of the offset vector, so the penetration depth between the shaft and bearing can be expressed as: S= e - c. (3) According to Eq. (3), the movement state of shaft and bearing can be expressed by 8, when 8 > 0, the shaft and bearing are in collision state, and 55 Dynamic Analysis of Hydraulic Support with Single Clearance Strojniski vestnik - Journal of Mechanical Engineering 67(2021)1-2, 53-66 the penetration depth is S; When S = 0, the shaft and bearing are in a state of contact or separation. When S < 0, the shaft and bearing are in a separate state; they both move freely without contact. 1.2 Clearance Contact Force Model Contact force will be generated during the collision between shaft and bearing and can be simply decomposed into normal force and tangential force. The direction of the normal force is the normal direction of the contact surface, mainly the normal collision force between the shaft and bearing, while the tangential force is the tangent direction of the contact surface, mainly the tangential friction between the shaft and bearing. After years of research, various researchers have proposed various models of normal contact force based on Hertz contact models. The Lankarani-Nikravesh contact force model takes into account the energy dissipation in the collision process and becomes a widely used nonlinear spring damping model in clearance research. According to the Lankarani-Nikravesh contact force model, the normal contact force Fn in the contact process can be expressed as: In Eq. (5), h1 and h2 are the material coefficients of the contact object and the equation is Eq. (6), u1 and u2 are Poisson's ratio of the contact material, E1 and E2 are the elastic modulus of the contact material. According to the different forms of friction, the tangential contact force model can be divided into a dry friction model and a lubrication friction model. The friction between the rear linkage and base belongs to dry friction, and the dry friction is generally modelled by the Coulomb friction model or the nonlinear friction model developed by domestic and foreign scholars. The nonlinear friction model proposed by Ambrosio based on the improvement of the Coulomb friction model is widely used, and its expression is as follows: F Cd Fn t. (7) In Eq. (7), is the sliding friction coefficient; Fn is the normal contact force; vT is the tangential relative velocity between the shaft and bearing; t is the direction of tangential contact force; cd is the dynamic correction coefficient, which is expressed as: F =K8"+Dv. (4) 0 vT - V V -V, 1 VT ^ Vs VT ^ Vd (8) In Eq. (4), S is the penetration depth between shaft and bearing; n is the force index, and n = 1.5 for the contact between metals; D is the damping coefficient of contact process; v is the normal collision speed between shaft and bearing; K is the stiffness coefficient of the contact process, and its equation is [32]: 4 ' — (5) K = 3n(hi + h2 ))R + R. h = i - u nE. i = 1,2. 54 (6) Zeng Q.L. - Li, Y.Y. - Yang, Y. In Eq. (8), vs and vd are specific velocity limit values. 2 DYNAMIC ANALYSIS OF HYDRAULIC SUPPORT WITH SINGLE CLEARANCE The key to the dynamic analysis of the hydraulic support with clearance between the rear linkage and base is to establish the clearance contact force model and simulate the hydraulic support model. The contact force model of the single clearance between the rear linkage and base can be described by the Lancarani- Vs