Simulation and Experimental Study on Indentation Rolling Resistance of a Belt Conveyor

Abstract

In this paper, in order to reduce the indentation rolling resistance of a belt conveyor and improve its operation efficiency, the indentation rolling resistance of a belt conveyor is studied. Firstly, the theoretical calculation of the indentation rolling resistance caused by the contact between a conveyor belt and an idler is studied. Additionally, the expression of the indentation rolling resistance including the multi-element Maxwell relaxation modulus is simplified and deduced using the assumption of small deformation, and the steps of iterative calculation of the indentation rolling resistance through setting the initial values are designed. Then, the multi-component Maxwell relaxation modulus function of the conveyor belt tested in the previous work of the authors is substituted into COMSOL 5.6 finite element software, and the stress distribution caused by the contact between the conveyor belt and the idler under different working conditions is analyzed. Also, a series of measures to reduce the indentation rolling resistance are found through the laws presented by the simulation results. Finally, the experimental study of the indentation rolling resistance is carried out using the indentation rolling resistance testing device from our research group. By comparing the experimental results for the indentation rolling resistance under different working conditions with the theoretical calculation results for the indentation rolling resistance, it was found that the relative error was less than 13%, which proved the feasibility of the indentation rolling resistance and relaxation modulus testing method, as well as the theoretical calculation method for indentation rolling resistance.

1. Introduction

The level of energy exploitation determines a country’s comprehensive national strength. In China’s industrial development and production, coal is the main energy source used, and the amount of coal used in China accounts for about 70% of all energy consumed [1]. Although in recent years, due to the impact of the pandemic, the total coal production in China has been somewhat reduced compared to before the pandemic [2], coal still plays an important role in China’s economic strength. Regarding mining machinery, large-scale transport devices are mainly used to transport coal after it is mined, and belt conveyors are the most used large-scale transport devices for transporting coal [3]. Because a belt conveyor can transport a large amount of coal a long distance all at once, its structure is not complicated, it can be easily maintained when it breaks down, it can be remotely controlled and dispatched using automation technology, and its operation efficiency is very high [4]. In the operation of all machinery in coal mines, the output power of a belt conveyor accounts for one-third of the total output power, but in the total energy consumption of all the machinery in a coal mine, a belt conveyor accounts for 60% [5].

In order to reduce the energy loss of a belt conveyor, it is necessary to analyze the friction force suffered by the belt conveyor in operation [6,7,8]. If the friction force is reduced, the energy consumption will also be reduced, and the working efficiency will naturally be improved. A belt conveyor has a series of friction forces: when the control system redirects the conveyor belt, it will produce redirection friction; there is friction force created through the process of placing coal by a conveyor belt feeding device; if the distribution of coal on the conveyor belt is loose, it will produce loose coal deformation friction; erroneous installation of the idler will also produce friction; the bending of the conveyor belt produces friction; the sweeper has friction; the rotation of the idler produces friction; and the indentation rolling resistance caused by the contact between the conveyor belt and the idler produces friction [9].

However, the indentation rolling resistance accounts for the most friction of all the friction forces, accounting for more than 60% of all the friction forces, so reducing the indentation rolling resistance can greatly reduce the running friction forces of a conveyor belt and improve its working efficiency [10,11,12].

A conveyor belt is composed of three layers [13]. The first layer and the third layer are called the upper and lower covering layers, respectively, and the main material used is rubber. The second layer is the belt core. The upper covering layer executes coal transportation and bears the weight of the coal, and the lower covering layer is driven by the idler to transfer the weight of the coal to the idler [14]. When the conveyor belt of a belt conveyor runs forward, it is the rotation of the idler that drives the conveyor belt to move forward to carry out coal mine transportation, so the idler will constantly come into contact with different parts of the rubber lower covering layer of the conveyor belt. When the roller breaks contact with the old part of the rubber lower covering layer of the conveyor belt, it will also come into contact with the new part of the rubber lower covering layer of the conveyor belt. If the rubber lower covering layer is made of strictly elastic material, as long as the old part breaks contact with the roller, its contact stress will disappear instantly, but the rubber is viscoelastic. When it breaks contact with the roller, the disappearance of the contact stress requires a process, that is, there is residual stress. The residual stress makes the resultant force of the area where the rubber lower covering layer of the conveyor belt and the idler are about to separate greater than that of the area where the rubber lower covering layer of the conveyor belt and the idler begin to make contact, and a moment is generated, which will hinder the movement of the conveyor belt. This is called the indentation rolling resistance moment [15,16,17,18]. In order to reduce the indentation rolling resistance, it is necessary to carry out a simulation and experiment research on the indentation rolling resistance under different working conditions.

2. Related Studies

Foreign researchers regard the contact between an idler and a conveyor belt as the contact between elastic material and viscoelastic material in the study of indentation rolling resistance. Wheeler et al. [19] used computer-aided engineering simulation software to study the changes in indentation rolling resistance under different working conditions, and Wheeler et al. [20] also used the Australian high-precision indentation rolling resistance experimental device to test the indentation rolling resistance of a belt conveyor, which also proved the correctness of the indentation rolling resistance simulation. Wheeler et al. [21] also proved that the materials of conveyor belts are different, and the indentation rolling resistance produced by them and the idlers is also different. Qin et al. [22] used computer-aided engineering simulation software to simulate and analyze the stress distribution caused by the mutual contact between an idler and a conveyor belt. Hötte et al. [23] determined that the original calculation method for the indentation rolling resistance between an idler made of tool steel and a conveyor belt needs to be improved. O’Shea et al. [24] also studied the test results of the indentation rolling resistance of a conveyor belt, and found that an inaccurate test of the viscoelastic parameters of the lower covering layer of a conveyor belt would lead to accumulated errors in the final test of the indentation rolling resistance.

In the domestic research on indentation rolling resistance under different working conditions, there is little research on the influence of the material of the conveyor belt on the indentation rolling resistance caused by the contact between the conveyor belt and the idler, and most of it focuses on the influence of the size of the conveyor belt and the idler and the running speed of the conveyor belt on the indentation rolling resistance. Li Guangbu [25] regarded the conveyor belt as an elastomer, and its relaxation modulus function was approximately equivalent to Young’s modulus when studying the indentation rolling resistance. Xu Furen [26] introduced the genetic integral operator based on Li Guangbu’s research to extend the Hertz formula to viscoelasticity and improved the formula for calculating the indentation rolling resistance of the conveyor belt. Tang Guoan et al. [27,28] used the Voigt model to characterize the conveyor belt in the study of indentation rolling resistance, in order to roughly calculate and analyze the indentation rolling resistance. Han Gang et al. [29] calculated the stress of the contact part between the conveyor belt and the idler and finally found that the distribution of the stress of the conveyor belt has a certain periodic change law.

To sum up the research status of the indentation rolling resistance at home and abroad, the formulas of the indentation rolling resistance are different because of the different models of the viscoelastic characteristics of the conveyor belt used by the researchers, the accuracy of the formulas of the indentation rolling resistance obtained by them is not high, and the influence of changing a certain working condition parameter on the indentation rolling resistance cannot be reflected. Compared with the Voigt model and the three-element Maxwell model, the multi-element Maxwell model is perfect for characterizing the characteristics of the conveyor belt. In this paper, the relaxation modulus function of the multi-element Maxwell model of conveyor belt obtained in previous work is used to carry out finite element simulation research on the indentation rolling resistance of conveyor belts, and the relaxation modulus function is used to calculate the indentation rolling resistance. Finally, the calculated value is compared with the experimental test value.

3. Study on Simplified Calculation of Indentation Rolling Resistance

Because the conveyor belt has three layers, but the one covering layer that makes contact with the idler will cause indentation rolling resistance, a simplified model can be established by combining the idler and conveyor belt, as shown in Figure 1 [30]:

The circle in Figure 1 represents the idler and the rectangle represents the conveyor belt. The conveyor belt transports to the right at a speed; the thickness of the conveyor belt is $h$; $t$ refers to the running time of the conveyor belt; $\delta (t)$ refers to the depression depth at the position of the conveyor belt with the abscissa $a-vt$, as shown in the figure; $h_0(t)$ is the depression depth at the position where the abscissa of the conveyor belt is 0, and it is also the maximum depression depth of the conveyor belt; $a$ is the abscissa corresponding to the point where the conveyor belt just comes into contact with the idler; $-b$ is the abscissa corresponding to the point where the conveyor belt is about to separate from the idler; $R$ is the radius of the idler.

Because $\delta (t)$ is very small compared with the radius $R$ of the idler, according to the circular plane geometry in Figure 1, the following expression is obtained:

$$\delta (t)={\displaystyle \frac{a^2-x^2}{2R}}$$

(1)

where $x$ represents the abscissa corresponding to the point at which the conveyor belt runs at time $t$.

By substituting $t=(a-x)/v$ into Equation (1), we can obtain the strain corresponding to the conveyor belt position at the moment t as follows:

$$\epsilon (t)={\displaystyle \frac{\delta (t)}{h}}={\displaystyle \frac{a^2-{(a-vt)}^2}{2Rh}}$$

(2)

The strain rate corresponding to the conveyor belt position at a time $t$ can be simplified as

$${\displaystyle \frac{d\epsilon (t)}{dt}}={\displaystyle \frac{v(a-vt)}{Rh}}$$

(3)

According to the stress formula of viscoelastic materials,

$$\begin{array}{l}\sigma (t)={\displaystyle {\int }_0^{t}E(t-\xi )\frac{d\epsilon (\xi )}{d\xi }d\xi }=\\ {\displaystyle {\int }_0^t(E_0+{\displaystyle \sum _{i=1}^N{E}_i{e}^{-\frac{t}{{\tau }_i}}{e}^{\frac{\xi }{{\tau }_i}}})\frac{v(a-v\xi )}{Rh}d\xi }\end{array}$$

(4)

where $t$ is the test time and the running time of the conveyor belt; $E(t)$ is the relaxation modulus function of conveyor belt rubber at time $t$; $N$ represents the number of Maxwell models; $E_i$ is the elastic modulus and the relaxation spectrum strength of the i-th spring in $2N+1$ of the generalized Maxwell model; ${\tau }_i$ is the ratio of the viscosity of the i-th sticky pot and the $E_i$ of the i-th spring, which is also known as the i-th relaxation time; $E_0$ is the value of E(t) when the time is infinite, that is, the equilibrium modulus. Since $t=(a-x)/v$, it can be obtained by substituting it into Equation (4):

$$\begin{array}{l}\sigma (x)={\displaystyle \frac{E_0{a}^2}{2Rh}}({\displaystyle \frac{a-x}{a}})({\displaystyle \frac{a+x}{a}})+{\displaystyle \sum _{i=1}^N\frac{E_i{a}^2{k}_i}{Rh}}[(1+k_i)\\ (1-e^{-{\displaystyle \frac{a-x}{k_{i}a}}})-{\displaystyle \frac{a-x}{a}}]\end{array}$$

(5)

where $k_i=v{\tau }_i/a$.

By integrating $\sigma (x)$ in Equation (5), the sum of the stresses on the conveyor belt, that is, the load $W$, can be obtained:

$$\begin{array}{l}W={\displaystyle {\int }_{-b}^a\sigma (x)}dx={\displaystyle \frac{a^3{E}_0}{6Rh}}(2+3\xi -{\xi }^3)+{\displaystyle \sum _{i=1}^N{E}_i}{k}_i\{{\displaystyle \frac{1}{2}}(1-{\xi }^2)-k_i[(1+k_i)\\ (1-e^{-(1+\xi )/k_i})-(1+\xi )]\}\end{array}$$

(6)

where $\xi =b/a$. The stress of the conveyor belt at $x=-b$ is 0, that is, $\sigma (-b)=0$; then,

$${\displaystyle \frac{E_0}{2}}(1+\xi )(1-\xi )+{\displaystyle \sum _{i=1}^N{E}_i}{k}_i[k_i-\xi -(1+k_i)e^{-(1+\xi )/k_i}]=0$$

(7)

The indentation rolling resistance of the conveyor belt can be obtained by continuous integration:

$$\begin{array}{l}F={\displaystyle \frac{{\displaystyle {\int }_{-b}^a\sigma (x)xdx}}{R}}={\displaystyle \frac{E_0{a}^4}{8R^{2}h}}(1-2{\xi }^2+{\xi }^4)+{\displaystyle \sum _{i=1}^N\frac{E_i{a}^4{k}_i}{R^{2}h}}[k_i^3-{\displaystyle \frac{k_i}{2}}(1+{\xi }^2)+{\displaystyle \frac{1}{3}}\\ (1+{\xi }^3)-k_i(1+k_i)(k_i+\xi )e^{-(1+\xi )/k_i}]\end{array}$$

(8)

From Equation (8), it can be seen that the indentation rolling resistance is related to the thickness, speed, and viscoelastic characteristics of the conveyor belt, radius of the idler, and load. When calculating the indentation rolling resistance of the conveyor belt, an iterative calculation can be conducted according to Equations (6)–(8). The flow chart of the indentation rolling resistance algorithm is shown in Figure 2.

Step 1: For a conveyor belt with a Maxwell model of 2N + 1, whose viscoelastic parameters are determined, a value of $\xi$ is initially given in Equation (6), which is recorded as ${\xi }_0$, and the radius value of the idler, the load value of the conveyor belt and the thickness value of the conveyor belt are substituted into Equation (6), and then $a$ can be solved in Equation (6).

Step 2: Substitute the solved $a$ into Equation (7), and solve $\xi$ in Equation (7) and record it as ${\xi }_1$. Compare the ${\xi }_0$ with the ${\xi }_1$, and if $\left|{\xi }_0-{\xi }_1\right|\le {10}^{-10}$, stop the iteration and directly enter the sixth step. If $\left|{\xi }_0-{\xi }_1\right|>{10}^{-10}$, proceed to the third step.

Step 3: Substitute ${\xi }_1$ into Equation (6), and other parameters of the conveyor belt and idler are also substituted into Equation (6), and then solve $a$.

Step 4: Substitute the solved $a$ into Equation (7) again, and solve the $\xi$ in Equation (7) and record it as ${\xi }_2$. Compare the ${\xi }_1$ with the ${\xi }_2$, if $\left|{\xi }_1-{\xi }_2\right|\le {10}^{-10}$, stop the iteration, and directly enter the sixth step. If $\left|{\xi }_1-{\xi }_2\right|>{10}^{-10}$, proceed to the fifth step.

Step 5: Continue iteration until $\left|{\xi }_{n-1}-{\xi }_n\right|\le {10}^{-10}$, where $n$ refers to the number of iterations.

Step 6: After the iteration, substitute the finally solved $a$, $\xi$ and other parameters of the conveyor belt and idler into Equation (8) to calculate the indentation rolling resistance of the belt conveyor under this specified working condition.

Therefore, for any given working condition, the above calculation flow can be used to calculate the indentation rolling resistance of the belt conveyor.

4. Finite Element Simulation Study on Indentation Rolling Resistance of Belt Conveyor Under Different Working Conditions

In this section, the finite element simulation research on the indentation rolling resistance of a belt conveyor under different working conditions is carried out, and the relaxation modulus function of the multi-element Maxwell model of belt rubber tested in previous work is used. The testing method and process of this relaxation modulus function can be found in reference [31] and they are not the focus of this paper, so this paper will not describe it in detail. COMSOL finite element simulation software is used for simulation. If the multi-body dynamics module of this software is used, multi-body dynamics can only simulate elastic materials, and the material parameters cannot be set to viscoelasticity. Therefore, the solid mechanics module is adopted to save the simulation time, as shown in Figure 3, and the automatic mesh generation of three-dimensional models is used for simulation. Figure 3 adopts the Winkler model, and its rationality has been confirmed by Lu Yan [32].

In Figure 3, the elastic cylinder represents the idler, and the viscoelastic horizontal plane represents the conveyor belt. The element is a solid element. The constraint conditions are as follows: The circle centers of the two ends of the idler can only move in the $Z$ axis direction, and the $X$ coordinates and $Y$ coordinates are fixed. The $Z$ coordinate of the bottom surface of the conveyor belt is fixed. Set the conveyor belt to move in the horizontal direction at the speed $v$. The material properties of the idler are set to those of metal materials, with a Poisson’s ratio of 0.24, a density of 7300 kg/m³, and Young’s modulus of 206 GPa. Subsequent simulation work of different loads can change the gravity of the idler by changing the density of the idler, and then the load can be changed.

The material property of the conveyor belt is viscoelasticity. The relaxation modulus function of conveyor belt rubber obtained by previous tests is as follows:

$$\begin{array}{l}E(t)=1.4180\times {10}^6+1.5232\times {10}^{-1}{e}^{-1000000t}+2.6677\times {10}^{-13}{e}^{-100000t}\\ +2.2204\times {10}^{-14}{e}^{-10000t}+2.1560\times {10}^5{e}^{-1000t}+4.3938\times {10}^5{e}^{-100t}+\\ 4.8751\times {10}^5{e}^{-10t}+4.5315\times {10}^5{e}^{-t}+2.2975\times {10}^4{e}^{-0.1t}+\\ 3.3752\times {10}^{-14}{e}^{-0.01t}+1.3180\times {10}^6{e}^{-0.001t}+1.4073\times {10}^6{e}^{-0.0001t}\end{array}$$

(9)

The viscoelastic parameters of the conveyor belt set in the software are shown in

Table 1. The viscoelastic parameters of the conveyor belt.

Branch	Relaxation Strength (Pa)	Relaxation Time (s)
1	1.5232 × 10−1	0.000001
2	2.6677 × 10−13	0.00001
3	2.2204 × 10−14	0.0001
4	2.1560 × 105	0.001
5	4.3938 × 105	0.01
6	4.8751 × 105	0.1
7	4.5345 × 105	1
8	2.2975 × 104	10
9	3.3752 × 10−14	100
10	1.3180 × 106	1000
11	1.4073 × 106	10,000
12	1.4180 × 106	999,999

, where the relaxation time corresponding to $E_0$ is set to infinity.

The radius of the idler is set to 0.045 m in the software; the thickness of the conveyor belt is set to 0.002 m; the length of the idler is set to 1 m; the weight of the idler, that is, the load, is set to 500 N; set the speed of the conveyor belt to 2 m/s, 6 m/s, and 10 m/s, respectively. The stress distribution images of the contact area at three speeds calculated by the software are shown in Figure 4.

In Figure 4, the origin of the abscissa corresponds to the abscissa of the circle center of the idler, the conveyor belt moves horizontally to the left, $x=a$ is the abscissa of the initial contact point between the conveyor belt and the idler, and $x=-b$ is the abscissa of the separation point between the conveyor belt and the idler. From the three curves shown in the figure, the following laws can be summarized: Under constant conditions, as speed increases, the normal distribution becomes increasingly asymmetric relative to the roller’s circle center. Additionally, the maximum normal stress rises, while the value of $a+b$ decreases. This indicates that the higher speed reduces the contact area between the conveyor belt and the roller, but causes the roller to press deeper into the conveyor belt. Therefore, reducing the running speed of the conveyor belt can reduce the indentation rolling resistance of the belt conveyor.

The radius of the idler to 0.045 m; the thickness of the conveyor belt is set to 0.002 m; the length of the idler is set to 1 m; the speed of the conveyor belt is set to 2 m/s; set the weight of the idler, that is, the load, to 500 N, 1500 N, and 2500 N, respectively. The stress distribution images of the contact area under three kinds of loads calculated by software are shown in Figure 5.

According to Figure 5, the following laws can be obtained: the normal stress distribution images under three loads are roughly similar, that is, the symmetry of the normal stress distribution is almost the same. With all other conditions being constant, an increase in load results in a rise in the maximum normal stress. As such, the contact area between the idler and conveyor belt ($a+b$) expands, and the pressing depth of the idler increases as well. Thus, the influence law of different loads is simple, and reducing the load can reduce the indentation rolling resistance of the belt conveyor.

The weight of the idler, that is, the load, is set to 500 N; the radius of the idler is set to 0.045 m; the length of the idler is set to 1 m; the speed of the conveyor belt is set to 2 m/s; the thickness of the conveyor belt is set to 0.002 m, 0.006 m, and 0.01 m, respectively. The stress distribution images of the contact area under three conveyor belt thicknesses calculated by software are shown in Figure 6.

According to Figure 6, the following rules can be drawn: with other conditions being unchanged, the normal stress distribution of the conveyor belt thickness of 0.006 m is more symmetrical with respect to the circle center of the idler compared with that of the conveyor belt thickness of 0.002 m, its maximum normal stress decreases, and the increase of $a+b$ means that the contact area between the roller and the conveyor belt increases, which shows that increasing the thickness of the conveyor belt is helpful in reducing the indentation rolling resistance. However, compared with the conveyor belt thickness of 0.006 m, the maximum normal stress of the conveyor belt thickness of 0.01 m does not decrease much, and the symmetry of the normal stress distribution and the change in $a+b$ are not great, which shows that when the conveyor belt thickness increases to a certain extent, the indentation rolling resistance does not change much.

The thickness of the conveyor belt is set to 0.002 m; the weight of the idler, that is, the load, is set to 500 N; the length of the idler is set to 1 m; the speed of the conveyor belt is set to 2 m/s; the radius of the idler is set to 0.045 m, 0.075 m, and 0.125 m, respectively. The stress distribution images of the contact area under three kinds of idler radii calculated by the software are shown in Figure 7.

According to Figure 7, with all other conditions remaining constant, an increase in roller radius leads to a decrease in the maximum normal stress, while the symmetry of the normal stress distribution remains nearly unchanged. The increase in $a+b$ means that the contact area between the roller and conveyor belt increases. On the whole, increasing the radius of the roller is helpful to reduce the indentation rolling resistance.

To sum up, this section uses COMSOL finite element software to simulate the distribution of normal stress in the contact area between the idler and the conveyor belt under different working conditions. By analyzing the value of $a+b$, the maximum normal stress, and the symmetry of the distribution of normal stress relative to the circle center of the idler, some measures to reduce the indentation rolling resistance are found.

5. Experimental Study on Indentation Rolling Resistance of Belt Conveyor Under Different Working Conditions

In order to verify the accuracy of the relaxation modulus function of the multi-element Maxwell model of conveyor belt rubber tested in previous work, and the rationality of the calculation flow of the indentation rolling resistance in the Section 3 of this paper, this section uses the indentation rolling resistance testing device of our research group to carry out experimental research on the indentation rolling resistance under different working conditions. Figure 8 is a schematic diagram of the indentation rolling resistance testing device used in this section.

Principle of indentation rolling resistance test: As shown in Figure 8, power is input by controlling the rotation speed of the driving roller, and the rotation of the driving roller drives the conveyor belt to move horizontally. The weight is hung below the test roller to apply load to the test roller, and the test roller is placed on the support plate to simulate the materials with uniform load distribution transported on the conveyor belt. One side of the tension pressure sensor is fixed on the right bracket, and the other side is connected to the loading system. Because the circle center of the test roller can only move in the vertical direction, the horizontal movement of the conveyor belt drives the test roller to rotate. According to the balance principle of force and moment in theoretical mechanics, the friction resistance caused by the contact between the conveyor belt and the test roller is equal to the tension force of the tension pressure sensor. Connect the tension pressure sensor with the amplifier, and then connect the amplifier with the data acquisition card, so that the tension pressure data can be tested on the Labview 2018 software on the computer, and the friction resistance is equal to the sum of the indentation rolling resistance and the rotation friction of the idler, so it is necessary to use the previous data on the rotation friction measured by the research group. Subtract the measured rotation friction value from the tension value of the tension pressure sensor to obtain the indentation rolling resistance value of the belt conveyor. The tightness adjustment system is connected to the reversing drum, and the tightness of the conveyor belt can be controlled by adjusting the height of the screw on the tightness adjustment system.

The physical diagram of the indentation rolling resistance testing device is shown in Figure 9:

The steps for testing indentation rolling resistance are as follows:

Step 1: Prepare for the test. Adjust the height of the screw on the tightness adjustment system to adjust the proper tightness of the conveyor belt, and make the calibrated tension pressure sensor as horizontal as possible with the ground.

Step 2: Load the conveyor belt. Load the conveyor belt with a weight, and properly vibrate the weight to make the test idler make full contact with the conveyor belt.

Step 3: Turn on the power supply and start the test. Let the conveyor belt move for a period of time, and after the movement is stable, start the Labview reading program on the computer to test the tension (average value of 15 s), and the tested tension is the friction generated by the contact between the conveyor belt and the test idler.

Step 4: Calculate the final indentation rolling resistance. Subtract the rotation friction measured by the research group from the measured tensile force, which is the value of the indentation rolling resistance caused by the contact between the conveyor belt and the test idler.

Use the experimental device and testing steps to carry out experimental testing on the indentation rolling resistance. The model of the conveyor belt is EP100, the upper covering layer on the conveyor belt is 6 mm, and the material of the upper covering layer on the conveyor belt is mainly rubber. The radius of the test idler is 55 mm and 65 mm. Test data of the indentation rolling resistance of the test roller with a radius of 55 mm are shown in

Table 2. Experimental test value of indentation rolling resistance of 55 mm radius test roller (N).

	Speed	0.5 m/s	1 m/s	1.5 m/s	2 m/s	2.5 m/s
Load
664.8 N		0.6593	1.0023	1.1771	1.2605	2.4088
738.5 N		0.7297	1.1225	1.3392	1.4353	2.5522
814.8 N		0.8285	1.2631	1.5173	1.6342	2.8162
850.0 N		0.8497	1.3385	1.6008	1.7275	2.9233

and

Table 3. Experimental test value of indentation rolling resistance of 65 mm radius test roller (N).

	Speed	0.5 m/s	1 m/s	1.5 m/s	2 m/s	2.5 m/s
Load
798.5 N		0.8015	1.2664	1.4966	1.6512	2.8126
872.3 N		0.8939	1.3957	1.6636	1.8387	3.0308
948.5 N		0.9803	1.5444	1.8503	2.0506	3.2690
984.8 N		1.0006	1.6050	1.9408	2.1478	3.3876

for the test roller with a radius of 65 mm.

Next, the indentation rolling resistance is calculated by using the calculation flow of the Section 3 and substituting it into the relaxation modulus function of the conveyor belt rubber in the Section 4, and the data in

Table 4. Theoretical calculation value of indentation rolling resistance of 55 mm radius test roller (N).

	Speed	0.5 m/s	1 m/s	1.5 m/s	2 m/s	2.5 m/s
Load
664.8 N		0.6380	0.9516	1.0873	1.1370	2.1483
738.5 N		0.7147	1.0767	1.2394	1.3024	2.3194
814.8 N		0.7946	1.2081	1.4002	1.4782	2.5020
850.0 N		0.8337	1.2710	1.4776	1.5632	2.5904

and

Table 5. Theoretical calculation value of indentation rolling resistance of 65 mm radius test roller (N).

	Speed	0.5 m/s	1 m/s	1.5 m/s	2 m/s	2.5 m/s
Load
798.5 N		0.7916	1.2229	1.4354	1.5291	2.5629
872.3 N		0.8702	1.3547	1.5997	1.7115	2.7544
948.5 N		0.9518	1.4921	1.7720	1.9037	2.9570
984.8 N		0.9908	1.5580	1.8550	1.9965	3.0551

can be obtained.

Comparing Table 2 with Table 4, and Table 3 with Table 5, the relative error statistics of experimental test values and theoretical calculation values can be carried out. The statistical results are shown in

Table 6. Relative error between theoretical calculation value and experimental test value of indentation rolling resistance of 55 mm radius test roller.

	Speed	0.5 m/s	1 m/s	1.5 m/s	2 m/s	2.5 m/s
Load
664.8 N		3.333%	5.326%	8.256%	10.858%	12.125%
738.5 N		2.096%	4.252%	8.056%	10.203%	11.258%
814.8 N		4.265%	4.556%	8.364%	10.552%	12.558%
850.0 N		1.918%	5.311%	8.335%	10.510%	12.851%

and

Table 7. Relative error between theoretical calculation value and experimental test value of indentation rolling resistance of 65 mm radius test roller.

	Speed	0.5 m/s	1 m/s	1.5 m/s	2 m/s	2.5 m/s
Load
798.5 N		1.255%	3.554%	4.265%	7.988%	9.744%
872.3 N		2.721%	3.027%	3.996%	7.432%	10.036%
948.5 N		2.995%	3.506%	4.418%	7.715%	10.552%
984.8 N		0.991%	3.014%	4.626%	7.577%	10.885%

.

By analyzing the data on indentation rolling resistance in Table 2, Table 3, Table 4 and Table 5, and the relative error data of theoretical calculation values and experimental test values of indentation rolling resistance in Table 6 and Table 7, the following laws can be obtained:

• For the experimental test results and theoretical calculation results, other conditions are certain: the greater the load, the greater the indentation rolling resistance, and the greater the belt speed, the greater the indentation rolling resistance.
• For any load and idler radius, the experimental test results of the indentation rolling resistance are the closest to the theoretical calculation results at a low speed of the conveyor belt, but the difference between them is slightly larger at a high speed.

For the second law, the following reasons for the differences are obtained:

• In the previous test of the viscoelastic characteristics of the relaxation modulus of conveyor belt rubber, the test result of the relaxation modulus function of conveyor belt rubber has a little error due to the influence of mechanical vibration of the viscoelastic testing device.
• The stiffness of the device for measuring the indentation rolling resistance of the conveyor belt and idler is slightly lower, which increases the error of the indentation rolling resistance test due to the mechanical vibration of the device when the running speed of the conveyor belt increases.
• There is also a slight error in the test results of roller rotation friction.
• There is a small error in the theoretical calculation method of indentation rolling resistance.
• When the running speed of the conveyor belt increases, it becomes more tense, which is equivalent to increasing the weak load.

From all the data in Table 6 and Table 7, the theoretical calculation results are generally close to the actual experiment, and the maximum relative error is less than 13%. Therefore, the test method of the relaxation modulus of conveyor belt rubber in previous papers, the test method of indentation rolling resistance of the conveyor belt and idler, and the theoretical calculation method of indentation rolling resistance in this paper are feasible.

6. Conclusions

In this paper, we reduced the indentation rolling resistance of a belt conveyor and improved its operation efficiency, and the following conclusions can be drawn through the finite element simulation and experimental study of the indentation rolling resistance of the belt conveyor under different working conditions:

• With other conditions being constant, the greater the belt speed, the greater the maximum normal stress between the conveyor belt and the idler, the smaller the contact area, and the more asymmetric the contact stress distribution with respect to the circle center of the idler. Therefore, reducing the belt speed is conducive to reducing the indentation rolling resistance of the belt conveyor.
• With other conditions being constant, the greater the load, the greater the contact area between the conveyor belt and the idler, and the greater the maximum contact normal stress, and the symmetry of the contact stress distribution is almost unchanged. So, reducing the load is beneficial to reducing the indentation rolling resistance.
• With other conditions being constant, the greater the thickness of the conveyor belt covering layer, the larger the contact area between the conveyor belt and the idler, the more symmetrical the contact stress distribution, and the smaller the maximum normal stress. Therefore, increasing the thickness of the conveyor belt covering layer can effectively reduce the indentation rolling resistance, but there is a threshold beyond which further increases in thickness yield little change in indentation rolling resistance.
• With other conditions being constant, the larger the radius of the idler, the larger the contact area between the conveyor belt and the idler, and the smaller the maximum normal stress, and the distribution symmetry of the contact stress is almost unchanged. Thus, increasing the radius of the idler is conducive to reducing the indentation rolling resistance.