This paper uses the Kelvin-Voigt viscoelastic model to establish the continuous dynamic equations for tail hammer tension belt conveyors. The viscoelastic continuity equations are solved using the generalized coordinate method. We analyze various factors influencing longitudinal vibration of the belt conveyor by simulation and propose a control strategy to limit the vibration. The proposed approach and control strategy were verified by several experimental researches and cases. The proposed approach provides improved accuracy for dynamic design of belt conveyors.
Belt conveyors are an integrated transmission and carrying mechanism with length sometimes extending several thousand meters. In traditional design and analysis of belt conveyors, vibration and impact are usually ignored and only static design is considered. However, to ensure the safety of conveyor operation for this restricted analysis, designers must increase the safety factor, which increases production costs.
Many research groups have conducted dynamic analysis of large belt conveyors to reduce production cost and optimize conveyor performance [
In the early 1960s, the former Soviet Union began to study conveyor dynamics. However, due to the limited science and technology at that time, the starting characteristics for constant acceleration or AC motors were studied by using impulse principles and stress wave propagation in the conveyor belt on the basis of a simplified mechanic model [
Harrison, Robert, and James amongst others investigated starting and braking characteristics of steel wire rope core conveyors and lateral bending vibration of the conveyor belt. They also analyzed stress wave propagation speed in the conveyor belt based on the theory of elastic and stress waves and developed various relevant models [
Computer simulation of belt conveyor dynamic characteristics was presented by the Taiyuan University of Science and Technology using a Kelvin viscoelastic model, and belt conveyor stability was analyzed in terms of the transverse vibration [
Starting and braking curves, horizontal turning, and broken band detection of belt conveyors were studied in the Liaoning Technical University using finite element analysis [
Modern conveyor design methods were proposed on the basis of analysis of the vibration characteristics by the Shanghai Jiao Tong University and Shanghai Normal University [
However, previous and current researches have mainly focused on discrete conveyor models. Although viscoelasticity has been considered, it has usually been simplified as an elastic model, and the actual conveyor dynamic model has rarely been considered. The current paper establishes the dynamic equation for tail hammer tension belt conveyor model based on the Kelvin-Voigt viscoelastic model, adopting the generalized coordinate method to solve the viscoelastic continuity equation. Conveyor natural frequency and longitudinal vibration characteristics are analyzed to provide a more accurate method for conveyor dynamic design.
The Kelvin-Voigt model comprises a linear spring and damper in parallel. The model is applicable to simulate the stress response of viscoelastic material. However, when
Solving (
This paper considers the belt conveyor as a continuous body, and the conveyor dynamic behavior is described by partial differential equations. Figure Compared to longitudinal vibration of conveyor belt, the effect of transverse vibration is negligible. Conveyor belt shearing and bending stress is negligible. Conveyor belt length change caused by vertical variation is negligible. Transverse deformation caused by conveyor belt longitudinal tension is negligible.
When starting a belt conveyor, besides being subjected to static tension, the conveyor belt is also subjected to dynamic tension, which is influenced by the conveyor speed change. Suppose that, at time
From (
Ignoring conveyor running resistance, the dynamic equation for conveyor longitudinal vibration can be expressed as
Substituting (
Considering belt conveyor dynamic characteristics, the differential equation for viscoelastic longitudinal vibration, neglecting static displacement, is [
When
Equation (
Thus, the homogeneous boundary conditions are
Equation (
Using the method of separation of variables, (
Suppose
Let
The subscript
The dynamic equation for conveyor longitudinal vibration can be expressed as
From (
Assuming that the belt conveyor adopts rectangular acceleration to start,
Figure The longitudinal vibration dynamical equation under step input is typical of second-order oscillation, where the natural and damping frequencies are functions of the load ratio. Underdamped vibration occurs as the damping coefficient changes from 0 to 1. As damping increases, maximum overshoot and transient time decrease, peak time increases, and stability improves. From the performance index of second-order systems, maximum system overshoot is
Overshoot for different damping coefficients.
Damping coefficient ( |
0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 |
---|---|---|---|---|---|---|
Overshoot (%) | 100 | 11.47 | 4.86 | 0.59 | 0.01 | 0 |
Table
Using (
Maximum dynamic tension in the conveyor belt occurs at the drive drum and gradually reduces away from the drum. Dynamic tension fluctuation amplitude decreases with increasing
Using the heavy hammer tension belt conveyor as an example, the equivalent hammer weight varies with the level of tension. From
Conveyor belt tension decreases with increasing tension. The maximum dynamic tension at the meeting point of the drive drum was 119.7, 113.0, and 108.4 kN, respectively. Consequently, to ensure normal operation, increasing tension can effectively reduce the conveyor belt dynamic tension.
Transport capacity is the major factor that determines the required belt conveyor power. Generally, the designer determines the required power based on full load or the harshest conditions. Under normal operating conditions, the belt conveyor dynamic characteristics will usually be significantly different for different loadings. Figure
Dynamic tension at the meeting point of the drive drum under no, half, and full load conditions was 47.95, 76.29, and 98.35 kN, respectively. Thus, dynamic tension at the meeting point of the drive drum increases with increasing transport capacity, but the relationship is nonlinear. Because the natural vibration frequencies of the belt conveyor are different for different loads, dynamic tension under a given load cannot be deduced directly from dynamic tension under a different load.
There are three major belt tension methods: heavy hammer, constant force, and fixed winch. Figure
Figure
The dynamic acceleration and tension of the belt conveyor system were simulated for large capacity, high speed, and long distance transport. Table
Simulation parameters for the considered belt conveyor.
Parameter | Symbol and value |
---|---|
Carrying capacity |
|
Haul distance |
|
Belt width |
|
Stable operation speed |
|
Weight of carrying idlers |
|
Mounting distance |
|
Weight of return idlers |
|
Mounting distance |
|
Linear mass per unit length |
|
Cross-sectional area |
|
Elasticity modulus |
|
Rheological constant |
|
Initial tension |
|
Let
The different starting regimes become more similar and acceleration maximum acceleration reduces as time increases. Dynamic tension fluctuation also converges and reduces with increasing time.
The parabolic starting regime and increased starting time to 15–20 × fundamental vibration period effectively reduced fluctuation and maximum dynamic acceleration and tension, effectively limiting longitudinal vibration shock across the whole system.
As shown in Figures
When starting time
Acceleration of the carrying segment for the whole system is increasing, moving from the head drive drum to tail tensioning, but the dynamic tension of the bearing segment decreases during the process of closing the tail. Therefore, in addition to choosing the starting regime and controlling starting time, further strategies were required to limit dynamic tension and acceleration volatility and maximum. The auxiliary equipment can be used in the middle part of the system, such as driving the middle linear friction wheel, arranging rational position and spacing and number of idlers, and choosing appropriate drums, to further enhance control of longitudinal vibration shocks for the overall system.
The belt conveyor simulation, run at 2 m/s, starts for 10–15 × fundamental period. Using a parabolic starting regime, dynamic experimental data of tension and acceleration were collected, for the physical conveyor system described in Table
Experimental belt conveyor system parameters.
Parameter | Value |
---|---|
Haul distance | 30 m |
Belt width | 500 mm |
Maximum speed under steady operation | 2.5 m/s |
Angle | 2° |
Drive drum diameter | 500 mm |
Turnabout electric drum diameter | 500 m |
Carrying rollers | 35° grooving rollers, 1.2 m spacing |
Return rollers | Pinto idlers, 2.4 m spacing |
Motor | Y-160 M, 4-5 kW, 380 V |
Retarder | K8717.1-AE5 |
Multiloop disc brake | / |
Constant hydraulic automatic tensioning | / |
Variable frequency soft starter | / |
Figure
Figure
To collect suitable data during the experiment, the dynamic testing device was installed close to the drive drum of the tail. Yet testing points of dynamic tension are close to the head and drum of the tail; the devices in the dynamic simulation experiment of acceleration and tension are shown in Figures
The site map of acceleration test.
The site map of tension test.
Figures
The tension cylinder oil inlet of the hydraulic automatic constant tension device starts to supply oil to the former at 5 s, which makes belt tension approximately linearly increase to operationally required initial tension (6 kN). In the following 5–31 s, the belt starts to move, and the system starts, reaching the normal running speed (2 m/s). When the system is running smoothly, tension is small, with the automatic tension cylinder providing a relatively stable 4.3 kN. Belt tension at the tail is relatively stable, due to dynamic buffering of the hydraulic automatic constant tension device.
Figures
When belt conveyor starts for 10 × fundamental period, peak belt acceleration at the tail and tension at the head are 0.018 m/s2 and 6.29 kN, respectively. When the belt conveyor starts for 15 × fundamental period, peak belt acceleration at the tail and tension at the head are 0.075 m/s2 and 6.079 kN, respectively. Thus, tension and acceleration are smaller when starting for 10 × fundamental period than 15, and starting is more stable.
We can conclude the following: A general solution was presented to simplify the viscoelastic rod system into partial differential continuous longitudinal vibration equations for longitudinal vibration. Also, a general theoretical approach was provided to analyze similar systems. Based on the longitudinal vibration dynamic equations, belt conveyor time response characteristics were analyzed under different starting control strategies. It was shown that the difference between maximum acceleration and maximum acceleration response was less than 5% when starting time was more than 15 × fundamental vibration period, and it can effectively reduce longitudinal vibration. Simulation results were experimentally verified. The effects on belt conveyor vibration characteristics from damping, tension, carrying capacity, tension mode, and belt speed were investigated and provide a theoretical basis for design and manufacture of belt conveyors.
The authors declare that they have no conflicts of interest.
The research was supported by the New Century Excellent Talents (Grant no. NECT-12-1038), NSFC-Shanxi Coal Based Low Carbon Joint Fund Focused on Supporting Project (Grant no. U1510205), and Science and Technology Project of Shanxi Province (Grant no. 2015031006-2). The authors gratefully acknowledge the helpful discussions with the research group and colleagues of Taiyuan University of technology.