A coal mine hoisting system includes two parts, one is a constantlength cable system, and the other is a variablelength cable system. In this paper, the nonlinear dynamic modeling of a coal mine hoisting system is established through Hamilton’s principle. The nonlinear partial differential equations of the coal mine hoisting system are discretized into ordinary differential equations by the fourthorder Galerkin truncation. The nonlinear dynamic responses and four key kinematic and structural parameter analysis of the coal mine hoisting system in the acceleration phases, constant velocity phases, and deceleration phases are given. The results show that the axial vibration displacements of the constantlength cable are an order of magnitude smaller than that of the variablelength cable. The load has the greatest effect on the axial vibration displacement of the hoisting cable. Adversely, the speed has the least effect on the axial vibration displacement of the hoisting cable.
The shallow coal resources are depleted. The development and utilization of ultradeep coal resources are becoming more and more important in the future [
Many scholars have done some research on coal mine hoist [
There are few literatures on nonlinear vibration behavior of the coal mine hoist. The kinematic and dynamic model of the mine hoisting system is similar to the elevator system or the axially moving system [
Few researchers have researched the dynamic behavior and kinematic and structural parameter analysis of a mine hoisting system with constant length and variable length. The coal mine hoist is primarily subject to axial excitation of its parameters without being externally motivated. In this paper, the nonlinear vibration equations of coal mine hoisting system are obtained through Hamilton’s principle. The fourthorder Galerkin truncation is used to discretize the vibration equations of the coal mine hoisting system. The nonlinear vibration behaviors and key kinematic and parameter characteristics of the hoisting system during the lifting or lowering process are analyzed.
The rest of paper is organized as follows. In Section
A coal mine hoisting system consists of two parts, one is a constantlength cable system, and the other is a variablelength cable system. The constantlength cable system and the variablelength cable system are coupled at the sheave. The model is shown in Figure
The model of a coal mine hoisting system.
In this paper, the modeling is based on the assumptions as follows:
The hoisting cable is uniform and continuous
The Young’s modulus, mass per unit length, and crosssectional area of the hoisting cable remain invariant
The slip line during operation is ignored
The torsional and transverse vibration responses of the hoisting cable are ignored
The elastic displacement due to axial vibration is greatly smaller than the length of hoisting cable
The kinetic energy of the coal mine hoisting system is as follows:
The elastic potential energy of the coal mine hoisting system is as follows:
The static tension
The gravitational potential energy of coal mine hoisting system is as follows:
During the movement of coal mine hoist, the hoisting cable and the sheave have sliding friction. The virtual work of the coal mine hoisting system is as follows:
The initial boundary condition of the coal mine hoisting system is as follows:
Through Hamilton’s principle, the motion of coal mine hoisting system between initial time
According to Equation (
Equations (
The axial vibrational displacements of the constantlength cable and the variablelength cable are coupled at the sheave. Therefore, the axial vibration displacement relationship between the constantlength cable and the variablelength cable is as follows:
The boundary governing equation of the coal mine hoisting system at the sheave is given as follows:
The boundary governing equation of the coal mine hoisting system at conveyance is given as follows:
The purpose of discretization is to discretize the differential equations that are difficult to solve into ordinary differential equations that are easy to solve. In this paper, the spatial partial differential vibration equations are discretized into ordinary differential equations by fourthorder Galerkin truncation [
To facilitate the analysis of the model, an independent variable
The axial vibration displacements of the constantlength cable and variablelength cable are approximated as follows:
The trial functions
By the independent variable
The partial derivatives of the variablelength cable are as follows:
The discretized differential equations of the constantlength cable and variablelength cable in coal mine hoist are given as follows:
The discretized differential equations are solved to obtain generalized coordinates
According to the mathematical model of the coal mine hoisting system, the real effect of the typical structural parameters on the hoister is analyzed. The coal mine hoist parameters are shown in Table
The coal mine hoist parameters.
Parameters  Value 

Radius of the sheave 
2.45 
Inertia moment of the sheave 
1.5 × 10^{4} 
Crosssectional area of the hoisting cable 
1.9 × 10^{−3} 
Young’s modulus of the hoisting cable 
1.01 × 10^{11} 
Mass per unit length of the hoisting cable 
8.6 
Constant length of the constantlength cable 
80 
Friction coefficient between the cable and sheave 
0.1 
Contact length between the cable and sheave 
3.5 
Angle of the constantlength cable and the horizontal plane 

Gravity acceleration 
9.8 
The axial vibration behaviors of coal mine hoist are different from the lifting or lowering process due to the timevarying characteristics of the cable. In order to prevent misoperation, it is assumed that the load of the coal mine hoist is full in lifting and lowering. The hoisting pattern of the coal mine hoist is shown in Figure
The hoisting pattern of the coal mine hoist. (a) In lifting the hoist. (b) In lowering the hoist.
The kinematic parameters of the coal mine hoist are as follows: the coil mine depth is 1580 m, the maximum velocity is 15 m/s, the maximum acceleration is 0.75 m/s^{2}, the hoisting time is 120 s, and the conveyance mass is 4 × 10^{4} kg.
The axial vibration displacements of the hoisting cable are shown in Figures
Axial vibration displacements of the constantlength cable at the sheave. (a) In lifting the hoist. (b) In lowering the hoist.
Axial vibration displacements of the variablelength cable at conveyance. (a) In lifting the hoist. (b) In lowering the hoist.
The coal mine hoist develops in the direction of high speed, high stability, heavy load, and ultradeep. The effects of kinematic and structural parameters such as hoisting load, hoisting height, hoisting acceleration, and hoisting velocity on the axial vibration responses of hoisting cable are not negligible. In the three operating phases of acceleration, constant velocity, and deceleration, the axial vibration responses of the hoisting cable is different. In this paper, four typical parameters of hoisting load, hoisting depth, hoisting acceleration, and hoisting velocity are discussed. The effects of four parameters on maximum axial vibration displacement of the hoisting cable in acceleration phases, constant velocity phases, and deceleration phases are analyzed.
The maximum axial vibration displacements of the constantlength cable and the variablelength cable under different loads are shown in Figures
Maximum axial vibration displacements of the constantlength cable under different loads. (a) In lifting the hoist. (b) In lowering the hoist.
Maximum axial vibration displacements of the variablelength cable under different loads. (a) In lifting the hoist. (b) In lowering the hoist.
The maximum axial vibration displacements of the constantlength cable and the variablelength cable under different depths are shown in Figures
Maximum axial vibration displacements of the constantlength cable under different depths. (a) In lifting the hoist. (b) In lowering the hoist.
Maximum axial vibration displacements of the variablelength cable under different depths. (a) In lifting the hoist. (b) In lowering the hoist.
The maximum axial vibration displacements of the constantlength cable and variablelength cable under different accelerations are shown in Figures
Maximum axial vibration displacements of the constantlength cable under different accelerations. (a) In lifting the hoist. (b) In lowering the hoist.
Maximum axial vibration displacements of the variablelength cable under different accelerations. (a) In lifting the hoist; (b) In lowering the hoist.
The maximum axial vibration displacements of the constantlength cable and variablelength cable under different velocities are shown in Figures
Maximum axial vibration displacements of the constantlength cable under the different velocities. (a) In lifting the hoist; (b) In lowering the hoist.
Maximum axial vibration displacements of the variablelength cable under different velocities. (a) In lifting the hoist. (b) In lowering the hoist.
It can be seen from Figures
In this paper, the mathematical equations of the coal mine hoisting system are established by Hamilton’s principle. The nonlinear vibration behaviors and key kinematic and structural parameters influences of the hoisting system are analyzed when the coal mine hoist is lifting or lowering. The results are shown as follows:
From acceleration phases to constant velocity phases and from constant velocity phases to deceleration phases, the axial vibration displacements of the hoisting cable changes significantly. The maximum axial vibration displacement of the constantlength cable is an order of magnitude smaller than the variablelength cable. The maximum axial vibration displacement of the hoisting cable in lifting process is similar to the lowering process.
In the acceleration phases, the axial vibration displacements have a linear relationship with the structural parameters. In the deceleration phases, the axial vibration displacements have a nonlinear relationship with the structural parameters. In the constant velocity phases, the different parameters and the hoisting process need to be analyzed.
The load change has the greatest effect on the maximum axial vibration displacements of the hoisting cable. The velocity change has the least influence on the maximum axial vibration displacements of the hoisting cable. The influence of depth and acceleration change on the maximum axial vibration displacements of the hoisting cable are less than the load change and greater than the velocity change. During the operation of the coal mine hoist, the overload operation should be prohibited. To increase the working efficiency of the hoist, the running velocity can be appropriately increased.
The data used to support the findings of this study are available from the corresponding authors.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was funded by the National Natural Science Foundation of China (Grant nos. 51675520, 51975569), and funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), Shandong Provincial Natural Science Foundation (Grant no. ZR2017QF011) and Weifang City Science and Technology Development Program (Grant no. 2017GX017).