1. Introduction
Belt conveyors are important transport equipment for transmitting bulk materials. Compared with heavy-duty trucks and trains, belt conveyors have some specific advantages such as large transport capacity, continuous operation, and low transport costs. More than two million conveyors are in operation annually in the world. Belt conveyors with long distances and heavy loads are widely used in coal mines, metallurgical industries, and other industries where the use of trucks and trains is impractical [
1,
2,
3]. Conveyor belts are made of rubber, steel wire, and fiber, which have some viscoelastic properties that give the belt conveyor complex dynamical characteristics. Serious accidents can take place during startup if the belt acceleration is too large for a belt conveyor with long distance and high power, such as slipping, tear, and severe wear [
4,
5,
6]. Speed control of the belt conveyors has become the key point of the medium-scale belt conveyors during starting and stopping [
7,
8,
9,
10,
11].
Soft-starting is required during startup for belt conveyors considering no slipping and no tearing. The belt speed curve should meet the requirements of less speed and less acceleration variation per unit time for the purpose of softness. Therefore, the parameters of belt speed and acceleration should be controlled by means of equipment. Using soft-starting equipment can realize the belt conveyor’s soft starting. The soft-starting technologies are mainly AC variable frequency speed regulation technology and hydro-viscous variable speed regulation technology. For variable frequency speed regulation technology, the output speed of the three-phase asynchronous motor can be regulated by adjusting its frequency [
12,
13]. It requires clean surroundings and sufficient ventilation, leading to high maintenance costs. For hydro-viscous variable speed regulation technology, the output speed of the wet clutch can be regulated by adjusting hydraulic pressure [
14,
15]. Similarly, it also requires excellent cleanliness and good ventilation. Apparently, the above two soft-starting technologies cannot meet the requirements of dirty working conditions such as those in underground coal mines. In addition, their purchasing costs are very high. Thus, a soft-starting technology with low costs and high reliability is needed for the belt conveyors used in adverse environments such as underground coal mines.
In the last two decades, AMT has been widely used in heavy-duty vehicles for some advantages of large transmitted torque, high transmission efficiency, low purchasing costs, low maintenance costs, and longer service life [
16,
17,
18,
19,
20]. The AMT has a high transmission efficiency of more than 90% [
21,
22], while other soft-starting technologies cannot provide such a high transmission efficiency. With continuous study and design, the input torque of the AMT supplied by ZF Friedrichshafen AG can reach 3400 N·m, which can meet the requirements of the transmitted torque for the heavy-duty vehicles and medium-scale belt conveyors with 500 kW. The author has published papers using AMT as a soft starter of the belt conveyors since 2013 [
23,
24]. The AMT soft-starting system for belt conveyors mainly contains a three-phase asynchronous motor, an AMT, a reducer, and a belt conveyor.
Researchers have done a great deal of theoretical and experimental studies on the soft-starting acceleration or speed curves in the past four decades, for the purpose of decreasing the belt acceleration and the belt jerk during starting. In 1983, Harrison proposed a sine acceleration curve for the belt conveyors and obtained an S-type belt speed curve [
25,
26]. In 1987, Nordell et al. proposed a triangular acceleration curve as a soft-starting acceleration curve for belt conveyors [
27,
28]. In 1994, Singh proposed a soft-starting acceleration curve with a creep section [
29]. In 1998, Song proposed a trapezoidal acceleration curve as a soft-starting acceleration curve for belt conveyors [
30]. In 2000, Bardos proposed a parabolic acceleration curve as a soft-starting acceleration curve for belt conveyors [
31]. All those soft-starting acceleration curves can make the belt accelerate softly and result in the belt speed curve being S-shaped. Therefore, designing a soft-starting acceleration for an AMT as a soft starter is key in starting the medium-scale belt conveyors softly.
Given a belt acceleration, the needed AMT output shaft’s angular acceleration can be calculated by driveline system parameters between the AMT and the conveyor. For the purpose of controlling the belt acceleration according to the designed curve, the AMT output shaft’s angular acceleration should be controlled according to the corresponding belt acceleration curve. In practice, the speed sensors measuring the AMT input and output shafts can provide accurate information when the AMT is working. In addition, the angular acceleration of the AMT input and output shafts can be calculated by the speed sensors. Therefore, controlling the AMT output shaft’s angular acceleration is possible. The paper provides a segmented belt acceleration curve for an AMT with many gears, which is a new belt acceleration curve in comparison with traditional belt acceleration curves.
The paper is organized as follows: In
Section 2, a segmented acceleration curve is proposed for the AMT as a soft starter. In
Section 3, the AMT soft starting system is described and the dynamic model is built. In
Section 4, the simulation model of the AMT soft-starting system based on AMESim software is built and the simulation results are analyzed.
2. Segmented Belt Acceleration Curve
The expressions of the four soft-starting acceleration curves as mentioned above are analyzed below. By analyzing the characteristics of AMT, a new soft-starting acceleration curve suitable to start the belt conveyors for AMT as a soft starter is developed.
The sine acceleration and its jerk curves are expressed as Equation (1).
where
is the time,
is the belt target speed of the belt conveyor,
is the starting time of the belt conveyor, and
and
are the belt acceleration and the belt jerk.
The triangular acceleration and its jerk curves are expressed as Equation (2).
The trapezoidal acceleration and its jerk curves are expressed as Equation (3).
where
is a natural number greater than or equal to 4 and
is the ascent or descent acceleration stage of the trapezoidal acceleration curve expressed as
.
The parabolic acceleration and its jerk curves are expressed as Equation (4).
Under the conditions of the same starting time and same belt target speed, the above four soft-starting accelerations and their jerk curves are presented in
Figure 1.
Figure 1a gives the comparison results from the changes of the acceleration curves, and
Figure 1b gives the comparison results from the changes of the jerk curves.
When N is greater than or equal to 4, ranking the maximum values of the belt acceleration from large to small, triangular acceleration, sine acceleration, parabolic acceleration, and trapezoidal acceleration curves occupy the first, second, third, and last places, respectively. When N is equal to 4, ranking the maximum value of the belt jerk from large to small, parabolic acceleration, trapezoidal acceleration, sine acceleration, and triangular acceleration curves occupy the first, second, third, and last places, respectively. When N is greater than 4, ranking the maximum values of the belt jerk from large to small, trapezoidal acceleration, parabolic acceleration, sine acceleration, and triangular acceleration curves occupy the first, second, third, and last places, respectively. Among the jerk curves, the triangular acceleration and trapezoidal acceleration curves have sudden change phenomena. From the perspective of control difficulties, the sine acceleration and parabolic acceleration curves are complicated to control, making their acceleration algorithms more complex than those of the other two soft-starting acceleration curves.
The above soft-starting acceleration curves can effectively control the AC variable frequency motor and the hydro-viscous start transmission. An AMT as a soft starter needs to upshift gradually to accelerate the belt, and the above four soft-starting acceleration curves cannot directly be used. Power needs to be cut using the clutch during upshifting, and the soft-starting acceleration curve should be considered in this situation. The transmitted torque of the clutch is related to the force of the diaphragm spring, and its value is a third-order polynomial related to the big end displacement of the diaphragm [
32,
33]. While the automatic clutch actuator is equipped to control the small end displacement of the diaphragm spring through thrust bearing, the big end displacement of the diaphragm spring can be controlled on account of the lever principle. Thus, the belt acceleration can be controlled by the transmitted torque of the clutch, which can be controlled by the clutch actuator. Considering the complexity of the shifting control and the clutch control, the algorithm of the belt acceleration should not be too complicated. Therefore, the soft-starting acceleration curve should be designed practically.
The clutch should be first disengaged for shifting and be engaged finally after shifting, and the power flow is interrupted during this interval. Thus, the belt cannot be accelerated during shifting. As can be seen from
Figure 1, the maximum value of the trapezoidal acceleration curve is lower than the maximum values of the other three soft-starting curves under the condition of the same starting time. In addition, the horizontal line of the trapezoidal acceleration curve lowers control difficulty for the clutch. Therefore, to minimize the belt acceleration and belt jerk, the trapezoidal acceleration curve is chosen as the basic acceleration curve and
N is set equal to 4. A segmented acceleration curve is proposed for an AMT as a soft starter based on the above analysis, as shown in
Figure 2.
The segmented acceleration curve includes several trapezoidal acceleration curves determined by the maximum running speed, motor rated speed, and gear positions. , , , and are the acceleration times for the respective gear positions. , , , and are designed maximum accelerations under the conditions of first gear, second gear, third gear, and fourth gear, respectively. , , and are starting times for accelerations under second, third, and fourth gear positions. For the purpose of stretching the whole belt, the belt is kept running for longer than 5 s after the acceleration stage labeled under first gear is completed. Owing to the belt being too long, the end of the conveyor end runs behind the head of the conveyor head during the acceleration process. That is to say, the time difference value between and contains 5 s and shifting time for second gear. The belt is kept running for more than 2 s for the purpose of making up the speed loss between the belt head and the belt end after the acceleration stage is completed under second gear or higher gear. Then, the AMT is shifted to another higher gear position. In this way, the upshifting operation should be completed until the designed target belt speed is reached and the soft-starting process of the belt conveyor is finished. Generally, more than five gear positions are provided by a heavy-duty AMT, which can meet the needs for belt speed.
The segmented acceleration curve for a conveyor belt is expressed as Equation (5).
4. Simulation Analysis Based on AMESim
4.1. Simulation Model and Parameter Setting
To show the dynamic response of soft-starting system based on the designed soft-starting acceleration curve for belt conveyor, a simulation model based on AMESim software using AMT as the soft starter for the belt conveyor with 300 kW is built as shown in
Figure 8.
The belt conveyor is built according to Equations (10), (16), and (17) from
Section 3.3. The upper and lower belts are divided into 10 parts separately. AMT model is composed of two parts: a clutch model and a transmission model. The model of the clutch transmitted torque is built according to Equation (23) from
Section 3.4. The motor model is built according to Equation (24) from
Section 3.5.
Ignoring the transmission efficiency of the driveline, the main parameters of the soft-starting system are listed in
Table 2.
Given the motor’s rated speed of 1485 rpm, the designed belt speeds from first to eighth gear positions are 0.54, 0.73, 0.99, 1.32, 1.78, 2.38, 3.20, and 4.32 m/s, respectively. Correspondingly, the AMT output shaft’s speeds from first to eighth gear positions are 103.99, 139.83, 188.69, 252.98, 339.43, 455.52, 611.11, and 825.00 rpm, respectively.
Limited by the belt’s maximum acceleration of 0.3 m/s2, the belt’s designed maximum acceleration according to the segmented acceleration curve is designed to be less than 0.2 m/s2 under different gear positions in this paper. Therefore, the acceleration stage times for every gear position are 4, 4, 4, 4, 6, 6, 8, and 10 s successively. After the acceleration stage for the first gear position is finished, the run time is 5 s. The run time is 3 s after the acceleration stage under other gear positions.
Owing to the needs of shifting and accelerating, clutch disengaging, shifting to neutral gear, choosing gear, shifting to a higher gear, and clutch engaging should operate successively. The time for clutch disengaging is 0.2 s, including 0.1 s for transmitting torque and 0.1 s for no torque. The time for shifting to neutral gear is 0.1 s. The time for choosing gear is 0.1 s. The time for shifting to a higher gear is 0.2 s. The time for clutch engaging before belt accelerating is 0.3 s, including 0.1 s for no torque (to eliminate the gap between the release bearing and the diaphragm small end) and 0.2 s for transmitting torque up to a half-engagement point. The time for clutch engaging during the belt accelerating stage is determined by the segmented acceleration curve for the gear position.
4.2. Simulation Results
Based on the above parameters, a simulation of the soft-starting process for a belt conveyor with 300 kW was conducted. To show the characteristics of the belt, the conveyor, and the AMT during the soft-starting process, some parameter variation curves are given below. The print interval of the simulation results is 0.01 s.
Figure 9,
Figure 10 and
Figure 11 show the speed, acceleration, and jerk of the upper belt, respectively.
Figure 12 shows the belt tensions of the upper belt.
Figure 13 shows the tensions of the driving pulley of the conveyor.
Figure 14 shows the speed, acceleration, and jerk of the AMT output shaft.
By shifting from first to eighth gear and acceleration control, the belt speed increases to the target speed of 4.32 m/s gradually, as shown in
Figure 9a. The belt speed decreases because of larger load inertia during the clutch disengaging for shifting. By comparison, the belt speed of the rear part lags behind that of the front part in
Figure 9b, which explains the belt viscoelasticity. Therefore, several seconds are needed to make the whole belt run to ensure that the rear part of the belt reaches the target speed after the front part of the belt reaches the target speed under every gear position of the AMT.
The belt acceleration curve from first to eighth gear is shown in
Figure 10a. The belt maximum acceleration of every gear position is less than 0.2 m/s
2, which meets the requirements of soft starting. The belt acceleration vibrates because of the belt resistance force during the clutch disengaging for shifting. Basically, the belt is accelerated according to the designed segmented acceleration curve. Obviously, the vibration of the front part is not stronger than that of the rear part. The belt acceleration curve at the beginning stage of soft starting is quite consistent with the designed segmented acceleration curve in
Figure 10b. With the starting of the front part, the rear belt follows.
The belt jerk curve from first to eighth gear is shown in
Figure 11a, which shows that the belt jerk during the belt accelerating stage is quite smaller than that during the clutch disengaging. The belt jerk changes with the belt acceleration. The belt maximum jerk of the front part is obviously greater than that of the rear part during the clutch disengaging under the influence of the driveline. The belt maximum jerk is 0.337 m/s
3 at the moment of 22.15 s during the belt accelerating process except for the time of power interruption because of shifting. The belt jerk curve at the beginning of soft starting is shown in
Figure 11b, which shows that the belt jerk of the rear part changes with that of the front part.
The belt tension curve from first to eighth gear is shown in
Figure 12. The belt maximum tension of the front part is given to drive the whole belt; the maximum value is 130,086 N during the first gear position corresponding to the belt acceleration curve. The belt maximum tension of the rear part is only 56,739 N during the first gear position. The belt tensions of the front and rear parts are 84,033 N and 45,124 N, respectively. The tension difference between the front and rear parts is needed to overcome the belt resistance force of the upper belt.
The belt tension curve of the driving pulley from first to eighth gear is shown in
Figure 13. The belt tension of the tight edge varies with the belt acceleration curve derived from the transmitted torque of the driving pulley. The belt tension of the loose edge changes little because the loose edge of the driving pulley is near the tensioning pulley. The belt tensions of the tight and loose edges are 84,033 N and 33,214 N, respectively, after the acceleration stage is finished under the eighth gear position.
The AMT output shaft’s speed, angular acceleration, and angular jerk curves during the whole soft-starting process from first to eighth gear are shown in
Figure 14. The AMT output shaft’s speed increases with the increasing of the gear position gradually, which is consistent with the belt speed. It reaches the speed of 827.06 r/min, which is determined by the motor speed and the transmission ratio of the eighth gear position. The AMT output shaft’s angular acceleration curve changes almost in line with the designed segmented acceleration curve if the power interruption is ignored. That is to say, the belt acceleration can be controlled by controlling the AMT output shaft’s angular acceleration determined by the clutch transmitted torque. The AMT output shaft’s maximum angular jerk is 215.06 rad/s
3 at the moment of 30.04 s.
By comparing the speed curves and the acceleration curves between the belt and the AMT output shaft, it can be seen that their shapes are similar, which shows that controlling the AMT output shaft’s accelerating process will control the belt accelerating process. The soft-starting process of the belt conveyor can be shown from front to rear parts; the parameters curves of the whole belt, including the belt speed, belt acceleration, and belt jerk, are clearly manifested.