# Cosine-Trapezoidal Soft-Starting Control Strategy for a Belt Conveyor

Authors: Jianhua Ji; Changyun Miao; Xianguo Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

To improve the soft-starting performance of a belt conveyor, a cosine-trapezoidal soft-starting control strategy is proposed. With this strategy, the speed curve for soft-starting is S-shaped, continuous, and smooth. The acceleration and jerk are also continuous and smooth without discontinuities. The time-interval ratio of the horizontal and cosine curves can be changed via a single parameter setting, so that the strategy can be modified to meet different practical needs. In this paper, the soft-starting control strategy of cosine trapezoid is analyzed and simulated based on current international and domestic standards. The results show that compared with the classical soft-starting control strategy, the cosine-trapezoidal soft-starting control strategy enables the belt conveyor to start at its highest efficiency. Moreover, the energy savings are obvious, the safety performance is good, and the controllable performance is excellent.


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## Body

## 1. Introduction

Because the conveyor belt has a certain viscoelasticity, the belt conveyor belongs to the class of flexible bodies. Therefore, the characteristics should be taken in full account in designing a soft-starting control strategy for the belt conveyor, so as to avoid or reduce as far as possible any impact damage of the motor on the belt conveyor. In addition, it is necessary to pay attention to ensure that the starting-up stage of the belt conveyor maintains a high starting efficiency and low energy consumption.The ideal starting process of the belt conveyor should ensure a smooth continuous acceleration and its maximum value is small to reduce the jolting impact of the motor on the belt conveyor. That is, the maximum jerk must be kept within a reasonable range to ensure a soft impact of motor on the belt conveyor [1–6]. Hence the speed curve should be S-shaped and the output smooth [7–10]. At the same time, it is necessary to ensure high efficiency and low energy consumption at startup. Although some reports have analyzed the operation efficiency and energy consumption of the belt conveyor [11–14], analyzing the efficiency and energy at startup is seldom performed.The simplest starting mode of belt conveyor is equal acceleration soft-starting control strategy, which is not often used because of its many shortcomings.(1)vt=VeTt,0≤t≤T(2)at=VeT,0≤t≤T(3)jt=0,0<t<T∞,t=0-∞,t=TIn 1983, for the purpose of minimizing the transient stress in the belt, Dr. A. Harrison [15] analyzed the circulating elastic waves and long period mass-spring oscillations of the whole belt conveyor system which disturbed the belt speed and belt tension, obtained the wave equation and deduced sinusoidal acceleration control strategy of the belt conveyor.(4)vt=Ve21-cos⁡πtT,0≤t≤T(5)at=πVe2Tsin⁡πtT≈1.57VeTsin⁡πtT,0≤t≤T(6)jt=4.93VeT2cos⁡πtT,0≤t≤TIn 1987, Nordell [16] put forward the triangular acceleration control strategy of belt conveyor.(7)vt=2VeT2t2,0≤t≤T2Ve-1+4tT-2t2T2,T2≤t≤T(8)at=4VeT2t,0≤t≤T24VeT1-tT,T2≤t≤T(9)jt=4VeT2,0≤t≤T2-4VeT,T2≤t≤TIn addition, parabolic acceleration soft-starting control strategy is often used [17].(10)vt=Ve3t2T2-2t3T3,0≤t≤T(11)at=1.5VeT4tT-4t2T2,0≤t≤T(12)jt=6VeT21-2tT,0≤t≤TIn all expressions,v(t) represents the startup speed,a(t) is the startup acceleration,j(t) denotes the startup acceleration,T denotes the startup time, andVe is the rated speed.From (2) and (3), the acceleration of the equal acceleration control strategy is discontinuous (mutation) at the start and end of the startup and the jerk is infinite. Therefore, the impact is greater on starting up the belt conveyor, and so its application is little used. The other three soft-starting control strategies are however widely used. Therefore, these three strategies are compared with our proposed soft-starting control strategy presented below.Referred to here as the cosine-trapezoidal control strategy, the theoretical analysis and simulation experiments that were performed and present comparison results with the above-mentioned classic control strategies are described in this paper. It is also concluded that this cosine-trapezoidal soft-starting control strategy is better than the others regarding soft-starting time-consuming start time, startup energy consumption, safety and control.


## 2. Cosine-Trapezoidal Soft-Starting Control Strategy

WithT the startup time of the belt conveyor the cosine-trapezoidal soft-starting control strategy (Figure 1) is characterized by a maximum acceleration value amax in the fixed time interval (T1,T2) and cosine curves in time intervals (0,T1) and (T2,T); the fixed times satisfy(13)T1=T-T2T2-T1=2NT1,N=0,1,2,3,⋯In the formula above,N denotes the number of half cycles of the cosine curve. That is,(14)T1=T2N+2T2=2N+1T2N+2,N=0,1,2,3,⋯Therefore, the expression for acceleration is(15)at=amax1-cos⁡πT1t,0<t<T12amax,T1<t<T2amax1-cos⁡πT1t,T2<t<TIntegrating these expressions with respect tot yields the velocity expressions(16)vt=amaxt-T1πsin⁡πT1t,0<t<T1amax2t-T1,T1<t<T2amaxT2-T1+t-T1πsin⁡πT1t,T2<t<TDifferentiating (15) with respect tot yields the jerk expressions(17)jt=amaxπT1sin⁡πT1t,0<t<T10,T1<t<T2amaxπT1sin⁡πT1t,T2<t<TThe running speed of the belt conveyor should reach the rated speed of Ve whent=T. Hence, from (16), the condition is obtained:(18)amaxT2-T1+T-T1πsin⁡πT1T=Vewhich, on substituting (14), becomes(19)amax=VeT·N+12N+1=VeT12+14N+2The substitution of (14) and (19) into (15)–(17) provides expressions parametrized byN andT for position, speed, acceleration, and jerk of the soft-starting control strategy:(20)dt=N+1Ve2N+1t22T+Tcos⁡2N+2π/Tt2N+22π2,0<t<T1N+1Ve2N+1t2T-t2N+2+T22N+22-2T2N+22π2,T1<t<T2N+1Ve2N+1NtN+1+t22T+Tcos⁡2N+2π/Tt2N+22π2-NT2N+2-T2N+22π2,T2<t<T(21)vt=N+1Ve2N+1tT-sin⁡2N+2π/Tt2N+2π,0<t<T1N+1Ve2N+12tT-12N+2,T1<t<T2N+1Ve2N+1NN+1+tT-sin⁡2N+2π/Tt2N+2π,T2<t<T(22)at=N+12N+1VeT1-cos⁡2N+2πTt,0<t<T12N+22N+1VeT,T1<t<T2N+12N+1VeT1-cos⁡2N+2πTt,T2<t<T(23)jt=N+12N+2π2N+1VeT2sin⁡2N+2πTt,0<t<T10,T1<t<T2N+12N+2π2N+1VeT2sin⁡2N+2πTt,T2<t<TSettingT=50 s, Ve=5 m/s, andN=2, simulation curves for the position, speed, acceleration, and jerk using our control strategy were plotted (see Figures 2–5).Figure 1
Design of the cosine-trapezoidal control strategy.Figure 2
Position curve of the cosine-trapezoidal soft-starting control strategy.Figure 3
Speed curve of the cosine-trapezoidal soft-starting control strategy.Figure 4
Acceleration curve of the cosine-trapezoidal soft-starting control strategy.Figure 5
Jerk curve of the cosine-trapezoidal soft-starting control strategy.


## 3. Simulation and Performance Analysis

The rated speed, maximum acceleration, and power of the belt conveyor all conform with the domestic and international standards. Limited by space, this study only uses a set of recommended values for analysis; the procedures and results of other numerical analyses are similar to those given below.


### 3.1. Startup Time and Jerk Analysis

In terms of the maximum acceleration and rated speed, the startup times and jerks of the four soft-starting control strategies can be given (Tables1 and 2). Setting the rated speed Ve=5 m/s and maximum acceleration amax=0.2 m/s2 for the belt conveyor, plots for the four soft-starting control strategies provide a base for comparison (Figure 6) and in particular a comparison of our cosine-trapezoidal control strategy for various values ofN (Figure 7).Table 1
Startup times comparison.
Control strategy Sinusoidal Triangle Parabolic Cosine-trapezoidal Startup time 1.57 V e a m a x 2 V e a m a x 1.5 V e a m a x 2 N + 2 2 N + 1 V e a m a xTable 2
Jerks comparison.
Control strategy Sinusoidal Triangle Parabolic Cosine-trapezoidal Maximum value 2 a m a x 2 V e a m a x 2 V e 2.67 a m a x 2 V e 1.57 2 N + 1 a m a x 2 V e Maximum ramp rate ∞ ∞ ∞ 4.93 2 N + 1 2 a m a x 3 V e 2 Mutation t = 0 , T t = 0 , T 2 , T t = 0 , T NoneFigure 6
Startup times for the different soft-starting control strategies.Figure 7
Change in startup times for the cosine-trapezoidal soft-starting control strategy withN.Given the same maximum acceleration, the startup time is much shorter using the cosine-trapezoidal control strategy than for the other three classical strategies. Moreover, its startup efficiency is the highest.Comparing cosine-trapezoidal strategies with differentN (Figure 7), the startup time shortened and the startup efficiency increases gradually asN increases. The shortest startup time of the cosine-trapezoidal soft-starting control strategy is Ve/amax (i.e.,N large).From Table2 and Figure 5, it can be seen that the jerk of the other three classical soft-starting strategies has a mutation, so their maximum ramp rate of jerk can be infinite at mutation. While the jerk of the cosine-trapezoidal soft-starting control strategy has no mutation, so its ramp rate of jerk is continuous and smooth. In addition, the jerk of the other three classical soft-starting control strategies is always nonzero during the startup process. But the jerk of the cosine-trapezoidal soft-starting control strategy is zero in [T1,T2]. Therefore, the cosine-trapezoidal soft-starting control strategy is conducive to reducing the damage of the soft impact on the belt conveyor.


### 3.2. Acceleration Analysis

Given a fixed startup time and rated speed, the maximum acceleration for the different soft-starting control strategies is listed in Table3. Setting Ve=5 m/s andT=50s, values of the maximum acceleration were then plotted for comparison (Figure 8). The maximum change in acceleration of the cosine-trapezoidal soft-starting control strategy for different values ofN was also plotted (Figure 9).Table 3
Maximum acceleration comparison.
Control strategy Sinusoidal Triangle Parabolic Cosine-trapezoidal Maximum acceleration 1.57 V e T 2 V e T 1.5 V e T 2 N + 2 2 N + 1 V e TFigure 8
Acceleration curves for the different soft-starting control strategies.Figure 9
Change in acceleration for the cosine-trapezoidal soft-starting control strategy withN.Table3 and Figure 8 show that the startup times are the same and the maximum acceleration of the belt conveyor using the cosine-trapezoidal soft-starting control strategy is much lower than the three classical strategies. That is, the belt conveyor experiences by far a softer impact under a cosine-trapezoidal strategy than under the other three strategies.From Figure9, with a cosine-trapezoidal strategy, asN increases, themaximum acceleration diminishes gradually. In the limit ofN large,the maximum accelerationtends to Ve/T (see Table 2).


### 3.3. Energy Saving Analysis

In the startup process, the energy saving criterion for the belt conveyor relates to the energy consumption rather than power. When the belt conveyor is moving at a constant speed, the circular force of the conveyor roller of the belt conveyor is given as(24)FU=CFH+FS1+FS2+FStwhereC is an added coefficient of resistance, FH the main resistance (N), FS1 the main special resistance(N), FS2 an added special resistance (N), and FSt the inclined resistance (N).During startup, the force of inertia of the belt conveyor is(25)FA=mL+mDatwheremL is the equivalent mass (kg) of the moving parts (conveyor belt, material, and roller) performing linear motion andmD is the equivalent mass (kg) of the rotating parts (without the roller) undergoing linear motion.During the startup period, the power of the belt conveyor (kW) is(26)pt=FU+mL+mDatvt1000whereas its energy consumption (kJ) is(27)wt=∫0tpxdxSet the rated speed of the belt conveyor to Ve=5 m/s, the maximum acceleration to amax=0.2 m/s2, the circumferential driving force of the drive roller to FU=50000 N, the equivalent mass of the moving body of the unloaded belt conveyor converted into linear motion of the conveyor belt to mLK=40000 kg, the full-loaded belt conveyor to mLm=120000 kg, and mD=30000 kg. The energy consumption and power of the four soft-starting control strategies for these settings are listed in Table 4, and the change in the energy consumption and power of the cosine-trapezoidal soft-starting control strategy for small values ofN are listed in Table 5.Table 4
Energy consumption and power of the different soft-starting control strategies.
Control strategy Comparison content Energy consumption/kJ Power/kW No load Full load No load Full load Sinusoidal 5453.1 6078.1 138.9 154.9 Triangle 7125 8125 142.5 162.5 Parabolic 5562.5 6562.5 148.3 175 Cosine-trapezoidal (N=2) 4637.2 5636.7 154.6 187.9Table 5
Energy consumption and power of the cosine-trapezoidal soft-starting control strategy for smallN.
N Comparison content Energy consumption/kJ Power/kW No load Full load No load Full load 0 7430.9 8418.2 148.6 168.4 1 5075.6 6074.2 152.3 182.2 2 4637.2 5636.7 154.6 187.9 3 4452.8 5452.4 155.9 190.8 4 4351.0 5350.8 156.6 192.6From Table4, it can be found that when the maximum acceleration is fixed, the energy consumption of the cosine-trapezoidal strategy is the lowest among the four strategies: the no-load energy consumption is 85% for the sinusoidal, 65.1% for the triangle, and 83.4% for the parabola, whereas the full-loaded energy consumption is 92.7% for the sinusoidal, 69.4% for the triangle, and 85.9% for the parabola. The power is slightly higher for the cosine-trapezoidal strategy than for the other three strategies.From Table5, using the cosine-trapezoidal strategy, with differentN values, the energy consumption diminishes withN whereas the startup power increases slightly. The energy saving with the cosine-trapezoidal strategy is evident.


### 3.4. Parameter N Analysis

#### 3.4.1. Influence of the Change inN on the Startup Time

SettingVe=5 m/s and amax=0.2 m/s2 into the startup time expression of the cosine-trapezoidal soft- starting control strategy gives(28)T=2N+22N+1Veamax=50N+12N+1sThe startup time changes withN. From (28) and the plot (Figure 10), the startup time decreases withN. Hence, from the perspective of a quick startup and improved efficiency,N should be large.Figure 10
Change of startup time of the cosine-trapezoidal soft-starting control strategy withN.


#### 3.4.2. Influence of the Change inN on the Energy Consumption

SettingVe=5 m/s and amax=0.2 m/s2, the circumferential driving force of the drive roller is FU=50000 N. The equivalent mass of the moving body of the no-load belt conveyor converted into linear motion of the conveyor belt is mLK=40000 kg and of the full-load belt conveyor is mLm=120000 kg. The equivalent mass of the rotating parts of the belt conveyor converted into linear motion of the conveyor belt is mD=30000 kg. The energy consumptions under no-load and full-load are then(29)WNo-Load=21375N2+28500N+9189.962N+12-7125N+40002N+1+2218.75kJ(30)WFull-Load=24375N2+32500N+10414.5592N+12-8125N+50002N+1+2968.75kJDuring startup, the energy consumption of the belt conveyor under no-load and full-load changes withN as evident in Figures 11 and 12.Figure 11
Change in the no-load energy consumption for the cosine-trapezoidal soft-starting control strategy withN.Figure 12
Change in the full-load energy consumption for the cosine-trapezoidal soft-starting control strategy withN.Using (29) and (30), the plots of the energy consumption of the belt conveyor (Figures 11 and 12) show a decrease with increasingN. Hence, to reduce the energy consumption requiresN to be large.


#### 3.4.3. Influence of the Change inN on Acceleration

Table2 and Figure 8 indicate that the maximum acceleration of the belt conveyor decreases with increasingN. Therefore, to mitigate hard impacts from the motor on the belt conveyor,N needs to be large.


#### 3.4.4. Influence of the Change inN on Jerk

From (23), the maximum jerk is(31)jmax=1.572N+1amax2VeWith Ve=5 m/s and amax=0.2 m/s2, the maximum jerk is(32)jmax=1.572N+1amax2Ve=0.012562N+1and hence, during startup, it increases linearly withN (Figure 13).Figure 13
Change in the maximum jerk for the cosine-trapezoidal soft-starting control strategy withN.If the jerk is too large, it may generate a large soft impact on the soft belt conveyor, and therefore its value should not be too high. Hence, with the objective to reduce the soft impact of the belt conveyor,N needs to be small.


#### 3.4.5. Influence of the Change inN on the Power

From Table5, the power of the conveyor increases slightly asN increases and inevitably increases the current. Hence a smallN is better to limit the current.That is, with increasingN, the energy consumption at startup is reduced, the efficiency is increased, and the maximum acceleration of the belt conveyor is reduced. However, whenN>5, the effect on all three objectives obviously diminishes. If 1≤N≤5, the maximum jerk remains within a reasonable range, the power does not rise significantly, and the effect on all three objectives stated above is obvious. Hence, the indications suggest a range forN of 1–5.


## 3.1. Startup Time and Jerk Analysis

In terms of the maximum acceleration and rated speed, the startup times and jerks of the four soft-starting control strategies can be given (Tables1 and 2). Setting the rated speed Ve=5 m/s and maximum acceleration amax=0.2 m/s2 for the belt conveyor, plots for the four soft-starting control strategies provide a base for comparison (Figure 6) and in particular a comparison of our cosine-trapezoidal control strategy for various values ofN (Figure 7).Table 1
Startup times comparison.
Control strategy Sinusoidal Triangle Parabolic Cosine-trapezoidal Startup time 1.57 V e a m a x 2 V e a m a x 1.5 V e a m a x 2 N + 2 2 N + 1 V e a m a xTable 2
Jerks comparison.
Control strategy Sinusoidal Triangle Parabolic Cosine-trapezoidal Maximum value 2 a m a x 2 V e a m a x 2 V e 2.67 a m a x 2 V e 1.57 2 N + 1 a m a x 2 V e Maximum ramp rate ∞ ∞ ∞ 4.93 2 N + 1 2 a m a x 3 V e 2 Mutation t = 0 , T t = 0 , T 2 , T t = 0 , T NoneFigure 6
Startup times for the different soft-starting control strategies.Figure 7
Change in startup times for the cosine-trapezoidal soft-starting control strategy withN.Given the same maximum acceleration, the startup time is much shorter using the cosine-trapezoidal control strategy than for the other three classical strategies. Moreover, its startup efficiency is the highest.Comparing cosine-trapezoidal strategies with differentN (Figure 7), the startup time shortened and the startup efficiency increases gradually asN increases. The shortest startup time of the cosine-trapezoidal soft-starting control strategy is Ve/amax (i.e.,N large).From Table2 and Figure 5, it can be seen that the jerk of the other three classical soft-starting strategies has a mutation, so their maximum ramp rate of jerk can be infinite at mutation. While the jerk of the cosine-trapezoidal soft-starting control strategy has no mutation, so its ramp rate of jerk is continuous and smooth. In addition, the jerk of the other three classical soft-starting control strategies is always nonzero during the startup process. But the jerk of the cosine-trapezoidal soft-starting control strategy is zero in [T1,T2]. Therefore, the cosine-trapezoidal soft-starting control strategy is conducive to reducing the damage of the soft impact on the belt conveyor.


## 3.2. Acceleration Analysis

Given a fixed startup time and rated speed, the maximum acceleration for the different soft-starting control strategies is listed in Table3. Setting Ve=5 m/s andT=50s, values of the maximum acceleration were then plotted for comparison (Figure 8). The maximum change in acceleration of the cosine-trapezoidal soft-starting control strategy for different values ofN was also plotted (Figure 9).Table 3
Maximum acceleration comparison.
Control strategy Sinusoidal Triangle Parabolic Cosine-trapezoidal Maximum acceleration 1.57 V e T 2 V e T 1.5 V e T 2 N + 2 2 N + 1 V e TFigure 8
Acceleration curves for the different soft-starting control strategies.Figure 9
Change in acceleration for the cosine-trapezoidal soft-starting control strategy withN.Table3 and Figure 8 show that the startup times are the same and the maximum acceleration of the belt conveyor using the cosine-trapezoidal soft-starting control strategy is much lower than the three classical strategies. That is, the belt conveyor experiences by far a softer impact under a cosine-trapezoidal strategy than under the other three strategies.From Figure9, with a cosine-trapezoidal strategy, asN increases, themaximum acceleration diminishes gradually. In the limit ofN large,the maximum accelerationtends to Ve/T (see Table 2).


## 3.3. Energy Saving Analysis

In the startup process, the energy saving criterion for the belt conveyor relates to the energy consumption rather than power. When the belt conveyor is moving at a constant speed, the circular force of the conveyor roller of the belt conveyor is given as(24)FU=CFH+FS1+FS2+FStwhereC is an added coefficient of resistance, FH the main resistance (N), FS1 the main special resistance(N), FS2 an added special resistance (N), and FSt the inclined resistance (N).During startup, the force of inertia of the belt conveyor is(25)FA=mL+mDatwheremL is the equivalent mass (kg) of the moving parts (conveyor belt, material, and roller) performing linear motion andmD is the equivalent mass (kg) of the rotating parts (without the roller) undergoing linear motion.During the startup period, the power of the belt conveyor (kW) is(26)pt=FU+mL+mDatvt1000whereas its energy consumption (kJ) is(27)wt=∫0tpxdxSet the rated speed of the belt conveyor to Ve=5 m/s, the maximum acceleration to amax=0.2 m/s2, the circumferential driving force of the drive roller to FU=50000 N, the equivalent mass of the moving body of the unloaded belt conveyor converted into linear motion of the conveyor belt to mLK=40000 kg, the full-loaded belt conveyor to mLm=120000 kg, and mD=30000 kg. The energy consumption and power of the four soft-starting control strategies for these settings are listed in Table 4, and the change in the energy consumption and power of the cosine-trapezoidal soft-starting control strategy for small values ofN are listed in Table 5.Table 4
Energy consumption and power of the different soft-starting control strategies.
Control strategy Comparison content Energy consumption/kJ Power/kW No load Full load No load Full load Sinusoidal 5453.1 6078.1 138.9 154.9 Triangle 7125 8125 142.5 162.5 Parabolic 5562.5 6562.5 148.3 175 Cosine-trapezoidal (N=2) 4637.2 5636.7 154.6 187.9Table 5
Energy consumption and power of the cosine-trapezoidal soft-starting control strategy for smallN.
N Comparison content Energy consumption/kJ Power/kW No load Full load No load Full load 0 7430.9 8418.2 148.6 168.4 1 5075.6 6074.2 152.3 182.2 2 4637.2 5636.7 154.6 187.9 3 4452.8 5452.4 155.9 190.8 4 4351.0 5350.8 156.6 192.6From Table4, it can be found that when the maximum acceleration is fixed, the energy consumption of the cosine-trapezoidal strategy is the lowest among the four strategies: the no-load energy consumption is 85% for the sinusoidal, 65.1% for the triangle, and 83.4% for the parabola, whereas the full-loaded energy consumption is 92.7% for the sinusoidal, 69.4% for the triangle, and 85.9% for the parabola. The power is slightly higher for the cosine-trapezoidal strategy than for the other three strategies.From Table5, using the cosine-trapezoidal strategy, with differentN values, the energy consumption diminishes withN whereas the startup power increases slightly. The energy saving with the cosine-trapezoidal strategy is evident.


## 3.4. Parameter N Analysis

### 3.4.1. Influence of the Change inN on the Startup Time

SettingVe=5 m/s and amax=0.2 m/s2 into the startup time expression of the cosine-trapezoidal soft- starting control strategy gives(28)T=2N+22N+1Veamax=50N+12N+1sThe startup time changes withN. From (28) and the plot (Figure 10), the startup time decreases withN. Hence, from the perspective of a quick startup and improved efficiency,N should be large.Figure 10
Change of startup time of the cosine-trapezoidal soft-starting control strategy withN.


### 3.4.2. Influence of the Change inN on the Energy Consumption

SettingVe=5 m/s and amax=0.2 m/s2, the circumferential driving force of the drive roller is FU=50000 N. The equivalent mass of the moving body of the no-load belt conveyor converted into linear motion of the conveyor belt is mLK=40000 kg and of the full-load belt conveyor is mLm=120000 kg. The equivalent mass of the rotating parts of the belt conveyor converted into linear motion of the conveyor belt is mD=30000 kg. The energy consumptions under no-load and full-load are then(29)WNo-Load=21375N2+28500N+9189.962N+12-7125N+40002N+1+2218.75kJ(30)WFull-Load=24375N2+32500N+10414.5592N+12-8125N+50002N+1+2968.75kJDuring startup, the energy consumption of the belt conveyor under no-load and full-load changes withN as evident in Figures 11 and 12.Figure 11
Change in the no-load energy consumption for the cosine-trapezoidal soft-starting control strategy withN.Figure 12
Change in the full-load energy consumption for the cosine-trapezoidal soft-starting control strategy withN.Using (29) and (30), the plots of the energy consumption of the belt conveyor (Figures 11 and 12) show a decrease with increasingN. Hence, to reduce the energy consumption requiresN to be large.


### 3.4.3. Influence of the Change inN on Acceleration

Table2 and Figure 8 indicate that the maximum acceleration of the belt conveyor decreases with increasingN. Therefore, to mitigate hard impacts from the motor on the belt conveyor,N needs to be large.


### 3.4.4. Influence of the Change inN on Jerk

From (23), the maximum jerk is(31)jmax=1.572N+1amax2VeWith Ve=5 m/s and amax=0.2 m/s2, the maximum jerk is(32)jmax=1.572N+1amax2Ve=0.012562N+1and hence, during startup, it increases linearly withN (Figure 13).Figure 13
Change in the maximum jerk for the cosine-trapezoidal soft-starting control strategy withN.If the jerk is too large, it may generate a large soft impact on the soft belt conveyor, and therefore its value should not be too high. Hence, with the objective to reduce the soft impact of the belt conveyor,N needs to be small.


### 3.4.5. Influence of the Change inN on the Power

From Table5, the power of the conveyor increases slightly asN increases and inevitably increases the current. Hence a smallN is better to limit the current.That is, with increasingN, the energy consumption at startup is reduced, the efficiency is increased, and the maximum acceleration of the belt conveyor is reduced. However, whenN>5, the effect on all three objectives obviously diminishes. If 1≤N≤5, the maximum jerk remains within a reasonable range, the power does not rise significantly, and the effect on all three objectives stated above is obvious. Hence, the indications suggest a range forN of 1–5.


## 3.4.1. Influence of the Change inN on the Startup Time

SettingVe=5 m/s and amax=0.2 m/s2 into the startup time expression of the cosine-trapezoidal soft- starting control strategy gives(28)T=2N+22N+1Veamax=50N+12N+1sThe startup time changes withN. From (28) and the plot (Figure 10), the startup time decreases withN. Hence, from the perspective of a quick startup and improved efficiency,N should be large.Figure 10
Change of startup time of the cosine-trapezoidal soft-starting control strategy withN.


## 3.4.2. Influence of the Change inN on the Energy Consumption

SettingVe=5 m/s and amax=0.2 m/s2, the circumferential driving force of the drive roller is FU=50000 N. The equivalent mass of the moving body of the no-load belt conveyor converted into linear motion of the conveyor belt is mLK=40000 kg and of the full-load belt conveyor is mLm=120000 kg. The equivalent mass of the rotating parts of the belt conveyor converted into linear motion of the conveyor belt is mD=30000 kg. The energy consumptions under no-load and full-load are then(29)WNo-Load=21375N2+28500N+9189.962N+12-7125N+40002N+1+2218.75kJ(30)WFull-Load=24375N2+32500N+10414.5592N+12-8125N+50002N+1+2968.75kJDuring startup, the energy consumption of the belt conveyor under no-load and full-load changes withN as evident in Figures 11 and 12.Figure 11
Change in the no-load energy consumption for the cosine-trapezoidal soft-starting control strategy withN.Figure 12
Change in the full-load energy consumption for the cosine-trapezoidal soft-starting control strategy withN.Using (29) and (30), the plots of the energy consumption of the belt conveyor (Figures 11 and 12) show a decrease with increasingN. Hence, to reduce the energy consumption requiresN to be large.


## 3.4.3. Influence of the Change inN on Acceleration

Table2 and Figure 8 indicate that the maximum acceleration of the belt conveyor decreases with increasingN. Therefore, to mitigate hard impacts from the motor on the belt conveyor,N needs to be large.


## 3.4.4. Influence of the Change inN on Jerk

From (23), the maximum jerk is(31)jmax=1.572N+1amax2VeWith Ve=5 m/s and amax=0.2 m/s2, the maximum jerk is(32)jmax=1.572N+1amax2Ve=0.012562N+1and hence, during startup, it increases linearly withN (Figure 13).Figure 13
Change in the maximum jerk for the cosine-trapezoidal soft-starting control strategy withN.If the jerk is too large, it may generate a large soft impact on the soft belt conveyor, and therefore its value should not be too high. Hence, with the objective to reduce the soft impact of the belt conveyor,N needs to be small.


## 3.4.5. Influence of the Change inN on the Power

From Table5, the power of the conveyor increases slightly asN increases and inevitably increases the current. Hence a smallN is better to limit the current.That is, with increasingN, the energy consumption at startup is reduced, the efficiency is increased, and the maximum acceleration of the belt conveyor is reduced. However, whenN>5, the effect on all three objectives obviously diminishes. If 1≤N≤5, the maximum jerk remains within a reasonable range, the power does not rise significantly, and the effect on all three objectives stated above is obvious. Hence, the indications suggest a range forN of 1–5.


## 4. Conclusion

For the belt conveyor, this paper proposed a different soft-starting control strategy providing high efficiency and low energy consumption, as well as offering stable, controllable, and safe operating conditions. The speed curve of the cosine-trapezoidal soft-starting control strategy isS-shaped allowing for continuous and smooth transitions without mutation as for the acceleration curve. Compared with the classic soft-starting control strategies, the advantages of the cosine-trapezoidal soft-starting control strategy are as follows:(1) Enhance startup efficiency: the startup time is short, and it is further shortened asN increases.(2) Provide energy savings: the energy consumption is lowest of the control strategies and can be further reduced by increasingN.(3) Improve controllable performance: the startup time, acceleration, jerk, and energy consumption can be changed via the load adjustment parameterN.(4) Raise safety performance: the hard/soft impacts imparted by the motor to the belt conveyor are suppressed and controlled.



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*Source: 8164247-2019-03-24.xml*