Indentation rolling resistance based on a three-parameter Kelvin solid model

Abstract

The indentation rolling resistance of a belt conveyor with stable operation accounts for more than 60% of the total resistance. Thus, the effective reduction of indentation rolling resistance and its energy consumption have attracted considerable attention of engineers. Indentation rolling resistance is due to the viscoelasticity of conveyor belt. A three-parameter Kelvin solid model is used to characterize the viscoelasticity of materials. The parameters of the Kelvin solid model include two elastic moduli $E_{0}$ and $E_{1}$ and one viscous coefficient $η_{1}$ , and their fitting functions are obtained by completing a dynamic tensile test. A mathematical model of indentation rolling resistance is derived, on the basis of theoretical analysis, which is directly expressed by belt speed, idler radius, and cover layer thickness under the conveyor belt, and is indirectly expressed by normal load and temperature. The influence of five factors on indentation rolling resistance and the influence of combined conditions of any two factors on indentation rolling resistance are separately analyzed on the basis of the mathematical model. The theory in this study is accurate based on the consistency between the theoretical calculation and experimental results. In practical engineering, the operating conditions of belt conveyors are any combination of all factors. Various factors should be considered to reduce indentation rolling resistance.

Keywords

Belt conveyor three-parameter Kelvin solid model indentation rolling resistance viscoelasticity working condition

Introduction

A belt conveyor can be used to transport materials at a constant or varying speed from a loading location to an unloading location on a certain logistics route, which forms a continuous stream or a pulsating stream.¹ The resistance generated by the belt conveyor in operation includes main, additional, special, and lifting resistance, in which the main resistance is most dominant.^2,3 The main resistance consists of roller running resistance, the material collision resistance, repeated bending resistance of the conveyor belt, and indentation rolling resistance (IRR). Considerable studies have corroborated that IRR accounts for a large proportion. Thus, IRR should be considered in designing a belt conveyor or investigating resistance.^4,5

Considerable studies at home and abroad have affirmed that the IRR of belt conveyors is due to its viscoelasticity. Jonkers⁶ deduced a formula to calculate IRR based on fieldwork experience. Vieweg et al.⁷ and Qiu⁸ proposed a two-dimensional (2D) model to determine the pressure distribution between the conveyor belt and the idler. Hunter⁹ and Morland¹⁰ presented a complete two-dimensional model that considers shear deformation. However, they assumed that the cover layer is a viscoelastic half-space. O’Shea et al.’s¹¹ research can be used to predict the rolling resistance of indentation. Lauhoff¹² focused on the effect of belt speed on energy conservation. Various representative models have been established to characterize the viscoelasticity of conveyor belts. Gallagher¹³ established a formula to calculate IRR based on the Maxwell model. Ziegler and Kraft¹⁴ used the extended Maxwell model to establish the rolling resistance between the idler and conveyor belts and collected IRR data under different idler speeds. Lu et al.¹⁵ used the Maxwell model to represent the viscoelasticity of materials and theoretically analyzed and experimentally verified IRR. Chen et al.¹⁶ predicted the IRR of a st3500 steel cord conveyor belt under different working environment temperatures. Rudolphi and Reicks¹⁷ extended the three-parameter Maxwell model to the 2N + 1 Maxwell model to express the viscoelasticity of conveyor belts. Munzenberger and Wheeler¹⁸ extended the IRR from theoretical analysis to experimental research. Although the above research has carried out in-depth studies on the rolling resistance of indentation from different angles, they have not taken into account the impact of environmental temperature changes on the IRR in 12 months.

The dynamic loading experiments of cylindrical rubber made of conveyor belt cover materials at different ambient temperatures are conducted to obtain the expression of elastic moduli $E_{0}$ and $E_{1}$ and viscosity coefficient $η_{1}$ based on the three-element Kelvin solid model. Thereafter, the fitting functions of the three parameters are obtained, which can intuitively reflect the influence law of temperature. Subsequently, a mathematical model of IRR is derived. The effect of temperature on IRR is reflected on the basis of elastic moduli and viscous coefficient. The influence of various factors on IRR is analyzed and evaluated on the basis of the working conditions of different ambient temperatures, belt speed, normal load, roller radius, and backing material thickness. The test is completed to verify the theory.

Fitting function of elastic moduli and viscosity coefficient

The three-element Kelvin solid model, also known as the standard linear solid model, is composed of the Kelvin model in series with a spring (Figure 1). $E_{0}$ and $E_{1}$ are the moduli of elasticity, and $η_{1}$ is the viscosity coefficient.

Figure 1.

Three-parameter Kelvin solid model.

A dynamic compression test is performed to determine the influence of temperature on the viscoelastic properties of conveyor belt materials. The purpose of this test is to obtain the performance parameters of rubber (i.e. elastic moduli and viscous coefficient) and to pave the way for the later theoretical research. Of course, the rubber sample material used in this section is the same as the bottom cover material of the conveyor belt in the IRR test for the later comparison of results. Elastic modulus is an important performance parameter of engineering materials. From the macroscopic point of view, elastic modulus is a measure of the resistance of an object to deformation. From the microscopic point of view, it is a reflection of bonding strength among atoms, ions, or molecules. Therefore, it can be seen that although the specimen is a cylinder, and the rubber of the conveyor belt for the IRR test is a cuboid, the change of shape does not hinder the acquisition of the parameters for the bottom cover of belt.

The rubber material used in the test is the rubber coating of a universal conveyor belt. The basic mechanical properties include a hardness (HA) of 76, a Mooney viscosity of 118, a tensile strength of 23.4 MPa, and an elongation at break of 732%. The rubber cylindrical sample with a radius of 50 mm and a height of 100 mm is evaluated. The sample is subjected to sinusoidal displacement loading on a high and low temperature test machine at a frequency of 0.1 Hz.¹⁶

In the experiment, the displacement response function is expressed as follows

B = A_{0} + A \sin (ω t)

(1)

where initial displacement $A_{0}$ is 6 mm, A is the amplitude, which is 5 mm, and $ω$ is the angular frequency.

Before starting the dynamic displacement stimulation, the temperature of the temperature control box is adjusted to −20°C, –10°C, 0°C, 10°C, 20°C, and 40°C and is maintained for 3 h to ensure that the compound inside the moving rubber sample is measured at the test temperature.

Figure 2 depicts that the dynamic force displacement curve of rubber apparently decreases with the increase of temperature. Meanwhile, the hysteresis loop area surrounded by the force displacement curve is gradually reduced. Therefore, temperature has an effect on the constitutive model parameters of the rubber.

Figure 2.

Experimental curve at different temperatures.

The steady-state solution of the three-element Kelvin solid model under sinusoidal strain is

\begin{matrix} F = & q_{0} \frac{A_{0} S}{H} + \frac{A (q_{1} - q_{0} p_{1}) ω S}{H (1 + ω^{2} p_{1}^{2})} \cos (ω t) \\ + \frac{AS (p_{1} q_{1} ω^{2} + q_{0})}{H (1 + ω^{2} P_{1}^{2})} \sin (ω t) \end{matrix}

(2)

where F is the force under strain and $q_{0}$ , $q_{1}$ , and $p_{1}$ are the model parameters

p_{1} = \frac{η_{1}}{E_{0} + E_{1}}, q_{0} = \frac{E_{0} E_{1}}{E_{0} + E_{1}}, q_{3} = \frac{η_{1} E_{0}}{E_{0} + E_{1}} η_{2}

(3)

Order

\begin{matrix} c_{0} = q_{0} \frac{A_{0} S}{H}, c_{1} = \frac{A (q_{1} - q_{0} p_{1}) ω}{H (1 + ω^{2} p_{1}^{2})} S, \\ c_{2} = \frac{A (p_{1} q_{1} ω^{2} + q_{0})}{H (1 + ω^{2} P_{1}^{2})} S \end{matrix}

(4)

Equation (2) is transformed into

F = c_{0} + c_{1} * \cos (ω t) + c_{2} \sin (ω t)

(5)

Formula (5) is a first-order form of Fourier. Parameters $c_{0}$ , $c_{1}$ , and $c_{2}$ in formula (5) can be obtained by fitting the dynamic load displacement curve of rubber with Fourier series.

The parametric expressions of elastic moduli and viscous coefficient can be obtained as follows

E_{0} = \frac{H (c_{1}^{2} A_{0} + c_{2}^{2} A_{0} - A c_{0} c_{2})}{AS (c_{2} A_{0} - c_{0} A)}

(6)

E_{1} = \frac{c_{0} H (c_{1}^{2} A_{0} + c_{2}^{2} A_{0} - A c_{0} c_{2})}{S {(c_{2} A_{0} - c_{0} A)}^{2} + {Sc}_{1}^{2} A_{0}^{2}}

(7)

η_{1} = \frac{H {(c_{1}^{2} A_{0} + c_{2}^{2} A_{0} - c_{0} c_{2} A)}^{2}}{AS c_{1} ω ({(c_{2} A_{0} - c_{0} A)}^{2} + c_{1}^{2} A_{0}^{2})}

(8)

The values of elastic moduli and viscosity coefficient corresponding to different temperatures can be calculated on the basis of formulas (6)–(8) and Figure 3. Polynomials are used to fit the data to clearly show the trend

\begin{matrix} E_{0} = & 283.3 T^{3} - 1.862 \times 10^{4} \times T^{2} - 2.18 \times 10^{6} T \\ + 7.511 \times 10^{8} \end{matrix}

(9)

E_{1} = 7.663 T^{3} - 225.7 T^{2} - 1.271 \times 10^{4} T + 8.966 \times 10^{5}

(10)

η_{1} = 0.08147 T^{3} - 1.64 T^{2} - 180.9 T + 6924

(11)

Figure 3.

$c_{0}$ , $c_{1}$ , and $c_{2}$ under different temperatures.

The R-squared values of elastic modulus and viscosity coefficient curves are 0.9952, 0.9751, and 0.9977, which indicate that the three curves have high fitting accuracy.

Establishment of IRR mathematical model based on the three-element Kelvin solid model

Static compression analysis

The indentation state between the cover layer of the conveyor belt and flat idler is investigated. Figure 4 illustrates the geometric model. The geometric model is based on several hypotheses, which are expressed as follows: (1) the cover layer of the conveyor belt is isotropic under full thickness; (2) the lower cover layer thickness h is greater than that of depression deformation $δ$ ; (3) the weight of the conveyor belt and material acts on the idler in the vertical direction; and (4) the contact half-length $a_{0}$ is less than that of the roller radius R.¹⁴

Figure 4.

Geometric model of static indentation zone of conveyor belt.

Compression deformation $δ$ of the conveyor belt is expressed as follows

δ = \sqrt{R^{2} - x^{2}} - R + h_{0} \approx h_{0} - \frac{x^{2}}{2 R} \approx \frac{a_{0}^{2} - x^{2}}{2 R}

(12)

where $δ$ is the belt deformation, R is the roller radius, and $h_{0}$ is the maximum depth of the belt.

The strain in the contact area between the idler and the belt is expressed as follows

ε = \frac{δ}{h} = \frac{h_{0}}{h} \sin (\frac{π}{2} - β)

(13)

where h is the cover layer thickness under the conveyor belt, $β = (\sqrt{2} / a_{0}) x$ , and $- a_{0} < x < a_{0}$ .

The stress corresponding to equation (13) is expressed as

σ_{0} = \frac{h_{0} E_{0}}{h} \cos β

(14)

In line with the equilibrium condition in the vertical direction, the positive pressure between the idler and the cover layer in the contact area of unit width can be obtained as follows

F_{v} = \int_{- a_{0}}^{a_{0}} σ_{0} dx = \sqrt{2} \times \sin \sqrt{2} \times \frac{h_{0}}{h} a_{0} E_{0}

(15)

$h_{0} \approx a_{0}^{2} / 2 R$ is obtained on the basis of geometric relationship $a_{0}^{2} + (R - h_{0})^{2} = R^{2}$ in Figure 4 available. Given $\sin \sqrt{2} \approx 1$ , the positive pressure is transformed into

F_{v} = \frac{a_{0}^{3}}{\sqrt{2} Rh} E_{0}

(16)

The mathematical model of the half chord length of the contact area can be obtained

a_{0} = {(\frac{\sqrt{2} F_{v} Rh}{E_{0}})}^{\frac{1}{3}}

(17)

When the conveyor belt is in a static state, the length of the contact area between the idler and the belt is symmetrical along the center line of the idler, and IRR is not formed at this time. Static analysis is the basis of research on dynamic IRR.

Establishment of mathematical model of dynamic indentation resistance

The geometric model of IRR is shown in Figure 5 when belt conveyor is in operation at a constant speed. Figure 5 shows that the contact arc length of the conveyor belt and the idler is not symmetrical about the centerline of the idler due to the viscoelastic properties of the underlying cover layer.^4,19

Figure 5.

Geometric model of the dynamic indentation zone of the conveyor belt.

In the research of the rolling resistance analysis, more studies select three-parameter Maxwell model to characterize the viscoelasticity of the conveyor belt material.^{4,13,14,15,19} In this article, the three-component Kelvin solid model is selected. At this time, the compression deformation of the conveyor belt is expressed as follows

δ (x) \approx \frac{a^{2} - x^{2}}{2 R}

(18)

where a is the coordinate value at $x = a$ .

The absolute coordinate of point $x = a$ is $x = a - vt$ at any time, by substituting it into equation (18) and sorting. Deformation $δ (t)$ with time is obtained as follows

δ (t) \approx \frac{2 avt - {(vt)}^{2}}{2 R}

(19)

where v is the belt speed.

The dynamic strain at each point of contact between the conveyor belt cover and the idler is written as

ε (t) = \frac{δ (t)}{h} = \frac{v}{2 Rh} (2 at - v t^{2})

(20)

The relaxation function of the three-element Kelvin solid model is as follows

Y (t) = q_{0} + (\frac{q_{1}}{p_{1}} - q_{0}) e^{- t / p_{1}}

(21)

The relaxed constitutive equation of viscoelastic material²⁰ is expressed as

σ (t) = \int_{- \infty}^{t} Y (t - τ) d ε (τ)

(22)

In the indentation region, the indentation stress varying with time t is expressed as follows

\begin{matrix} σ (t) = & \frac{q_{0} v}{2 Rh} (2 at - v t^{2}) \\ + \frac{v}{Rh} (q_{1} - q_{0} p_{1}) [(a + v p_{1}) (1 - e^{- t / p_{1}}) - vt] \end{matrix}

(23)

Substituting $t = (a - x) / v$ in the above formula, the indentation stress at any point in the indentation area is the following

\begin{matrix} σ (x) = & \frac{q_{0}}{2 Rh} (a^{2} - x^{2}) \\ + \frac{v}{Rh} (q_{1} - q_{0} p_{1}) [(a + v p_{1}) (1 - e^{\frac{x - a}{v p_{1}}}) - a + x] \end{matrix}

(24)

In line with the equilibrium conditions in the vertical direction, the vertical load expression can be acquired as follows

\begin{matrix} P = \frac{q_{0}}{6 Rh} (2 a^{3} + 3 a^{2} b - b^{3}) + \frac{v}{Rh} (q_{1} - p_{1} q_{0}) \\ [\frac{a^{2} - b^{2}}{2} + v p_{1} (b - v p_{1}) + v p_{1} (a + v p_{1}) e^{\frac{- b - a}{v p_{1}}}] \end{matrix}

(25)

The resistance moment of the stress distribution on the conveyor belt to the center of the idler is expressed as follows

M = \int_{- b}^{a} x σ (x) dx

(26)

The IRR mathematical model is derived from equation (26) and $F_{z} = M / R$

\begin{matrix} F_{z} = & \frac{q_{0}}{8 R^{2} h} (a^{2} - b^{2})^{2} + \frac{v}{R^{2} H} (q_{1} - p_{1} q_{0}) \\ [{(v p_{1})}^{3} - \frac{1}{2} (a^{2} + b^{2}) v p_{1} + \frac{a^{3} + b^{3}}{3} \\ + v p_{1} (a + v p_{1}) (b + v p_{1}) e^{\frac{- b - a}{v p_{1}}}] \end{matrix}

(27)

Formula (27) is the IRR mathematical model, which is represented by the parameters $q_{0}$ , $q_{1}$ , $p_{1}$ . The model cannot visually see the impact of the elastic moduli and the viscosity coefficient on the IRR, so that the formula is sorted out.

Order $k = v τ / a$ , and $λ = b / a$ .

Then the resulting mathematical model of IRR is as follows

\begin{matrix} F_{z} = \frac{E_{0} E_{1}}{E_{0} + E_{1}} \cdot \frac{a^{4}}{8 R^{2} h} (1 + λ^{4} - 2 λ^{2}) \\ + \frac{{E_{0}}^{2} E_{1}}{{(E_{0} + E_{1})}^{2}} \cdot \frac{k a^{4}}{R^{2} h} [k^{3} \cdot {(\frac{E_{1}}{E_{0} + E_{1}})}^{3} - \frac{k}{2} (1 + λ^{2}) \cdot \frac{E_{1}}{E_{0} + E_{1}} \\ + \frac{1 + λ^{3}}{3}] + \frac{{E_{0}}^{2} {E_{1}}^{2}}{{(E_{0} + E_{1})}^{3}} \cdot \frac{k^{2} a^{4}}{R^{2} h} (1 + \frac{k E_{1}}{E_{0} + E_{1}}) \\ (λ + \frac{k E_{1}}{E_{0} + E_{1}}) e^{\frac{1 + λ}{- k} \cdot \frac{E_{0} + E_{1}}{E_{1}}} \end{matrix}

(28)

Analysis of theoretical results

Many factors affect IRR based on the above analysis. The direct factors include belt speed, idler radius, and cover layer thickness under the conveyor belt, and the indirect factors are vertical load and temperature. Figures 6 –10 show the analyses of the effect of a single factor on IRR. Figures 11 –20 show the change of IRR under the action of any two factors.

Figure 6.

Two-dimensional relationship between belt speed and IRR—P = 600 N/m, R = 44.5 mm, h = 3 mm, T = 0°C.

Figure 7.

Two-dimensional relationship between normal load and IRR—v = 2 m/s, R = 38 mm, h = 1.5 mm, T = –20°C.

Figure 8.

Two-dimensional relationship between idler radius and IRR—P = 1200 N/m, v = 4 m/s, h = 6 mm, T = 10°C.

Figure 9.

Two-dimensional relationship between bottom cover thickness and IRR—P = 1800 N/m, R = 54 mm, v = 6 m/s, T = 30°C.

Figure 10.

Two-dimensional relationship between working temperature and IRR—P = 2400 N/m, R = 66.5 mm, v = 8 m/s, h = 8°C.

Figure 11.

Three-dimensional relationship diagram of normal force, idler radius, and IRR—v = 1 m/s, h = 3 mm, T = 0°C.

Figure 12.

Three-dimensional relationship diagram of normal force, bottom cover thickness, and IRR—v = 2 m/s, R = 44.5 mm, T = 0°C.

Figure 13.

Three-dimensional relationship diagram of normal force, belt speed, and IRR—R = 65.5 mm, h = 5 mm, T = 10°C.

Figure 14.

Three-dimensional relationship diagram of working temperature, normal load, and IRR—v = 3 m/s, R = 44.5 mm, h = 6 mm.

Figure 15.

Three-dimensional relationship diagram of belt speed, idler radius, and IRR—P = 600 N/m, h = 3 mm, T = –20°C.

Figure 16.

Three-dimensional relationship diagram of belt speed, bottom cover thickness, and IRR—P = 1200 N/m, R = 54 mm, T = 20°C.

Figure 17.

Three-dimensional relationship diagram of belt speed, working temperature, and IRR—P = 1800 N/m, R = 44.5 mm, h = 6 mm.

Figure 18.

Three-dimensional relationship diagram of bottom cover thickness, idler radius, and IRR—P = 2400 N/m, v = 8 m/s, T = 30°C.

Figure 19.

Three-dimensional relationship diagram of idler radius, working temperature, and IRR—P = 3000 N/m, v = 6 m/s, h = 8 mm.

Figure 20.

Three-dimensional relationship diagram of working temperature, bottom cover thickness, and IRR—P = 1200 N/m, v = 10 m/s, R = 44.5 mm.

Generally, the belt conveyor runs at a speed of 0.1–6.5 m/s. However, the speed of 6.5 m/s cannot meet the engineering requirements due to the development of high-speed and long-distance belt conveyors. Thus, the belt speed ranges between 1 and 10 m/s. The loads for simulating material and conveyor belt weight are to be 600, 1200, 1800, 2400, and 3000 N/m. The thickness values of the backing cover layer are 1.5, 3, 5, 6, 8, and 10 mm. The ranges of idler radius are 38, 44.5, 54, 66.5, and 79.5 mm. The range of the above factors is determined on the basis of the design manual and the actual working conditions of the project.

Figures 6 –10 show 2D relationship diagram of belt speed, normal load, roller radius, backing cover thickness, temperature, and IRR. The belt speed and the IRR tend to increase or decrease together. The IRR slowly increases when the belt speed reaches 7 m/s. The increase of normal load leads to the steady increase of IRR, and the increase range is large. The IRR decreases with the increase of roller radius. On the contrary, the IRR increases with the increase of cover layer thickness. The influence of temperature on IRR is complicated. In the process of gradual warming, the IRR is slightly reduced, and the minimum value is obtained at 30°C. Subsequently, the temperature and IRR increase. The influence of various factors on IRR is not a simple linear.

Figures 11 –20 show the three-dimensional relationship between two factors and IRR. Each figure represents a working environment. The red color in the figure indicates a large value and it can be seen that variation law of IRR is different under the action of any two factors. Due to different working conditions, it is impossible to simply compare the numerical values of the IRR. However, the IRR is the smallest under small belt speed, large idler radius, small lower cover layer, and the high the temperature.

Test verification

The authors develop an IRR test system to verify the accuracy of the proposed theory (Figure 21). The test system is placed in the temperature control room shown in Figure 22 to facilitate the adjustment of test temperature.

Figure 21.

Indentation rolling resistance test rig.

Figure 22.

Test ambient temperature control room.

The test system can collect data under different roller radii, belt speeds, vertical loads, and different types of conveyor belt at different temperatures. Various sensors in the system are connected with the data collection system, and the data are directly transmitted to the computer for easy collection. During data acquisition, the operation of the belt conveyor can be viewed at any time through an observation window (shown Figure 23) in order to make adjustments on the test.

Figure 23.

Test observation window.

The test system consists of a driving pulley and a driven pulley with a diameter of 800 mm. There is a supporting drum in the middle of the conveyor belt, which has a diameter $D_{R . G}$ of 400 mm and a length of 500 mm. The wheelbase L is 3800. The diameter $D_{R . M}$ of the idler selected in this test is 76 mm, and the above dimensions all meet the requirements of DIN 22123:2012,²¹ and the size relationship also satisfies the following formula

D_{R . G} \geq 1.5 \times D_{R . M}

(29)

During the experiment, the belt speed can be digitized using a photoelectric sensor, and belt speed adjustment is performed using the frequency inverter. The weight is loaded by applying pressure on the hydraulic cylinder. Before each loading, the laboratory should to be adjusted to the expected temperature in advance, and the entire experimental system should be stationary at the temperature for 3 h to ensure that the characteristics of the entire test system match the temperature.

In this test, a fabric conveyor belt with a bandwidth of 400 mm and a thickness of 3 mm under the cover is used. The bandwidth meets the minimum bandwidth requirement of 3500 mm in DIN 22123:2012, and the ratio of support drum length to the test bandwidth is more than 1.1. Each test should be evaluated at least three times under different working conditions, and the final average value is got. The whole testing process is based on the requirements and regulations of DIN 22123:2012. The experimental data are compared with the theoretical results as shown in Figures 24 and 25.

Figure 24.

Comparison of theoretical and experimental results at different belt speeds—T = 10°C, R = 38 mm, h = 3 mm.

Figure 25.

Comparison of theoretical and test results at different loads—v = 2 m/s, R = 38 mm, h = 3 mm.

As can be seen from Figures 24 and 25, the experimental results are basically consistent with the theoretical results. However, several errors are found between the experimental and theoretical results. The experimental values are generally larger than those of the theoretical values, which is due to the influence of IRR and sensor in the test. The greater the IRR is, the greater the influence by external factors will be.

Conclusion

The study of IRR is based on the three-parameter Kelvin solid model. The elastic modulus and viscosity coefficient in the three-parameter Kelvin model can reflect the viscoelastic properties of the conveyor belt. The data obtained by dynamic tensile test are fitted to obtain a functional expression of the three parameters, which characterize the effect of temperature on the viscoelasticity of materials. On this basis, the IRR mathematical model is derived. Thus, belt speed, normal load, idler radius, the thickness of the bottom cover layer, and the temperature can affect the value of IRR. However, the influence law is not the same.

The increase of belt speed, normal load, and thickness of the bottom cover layer cause the increase of IRR. In comparison, the change caused by the normal load is larger. IRR appears to be stagnant when the belt speed exceeds 7 m/s. The backing cover layer has the least influence on IRR. The relationship between the idler radius and IRR is a trade-off relationship, that is, the idler radius is increased, and IRR is reduced. The influence of temperature on IRR is the most complex. When the temperature is lower than 30°C, the IRR decreases with increasing temperature. The IRR increases when the temperature increases to 40°C.

The test verification is completed on the IRR test system, which can adjust the normal load, belt speed, and temperature of test environment. The general trend is the same although a certain error is found between the theoretical and test results. In actual engineering, the consideration of only one factor is unacceptable when a company wants to achieve an energy-saving effect. The impact of all influencing factors should be comprehensively considered.

Footnotes

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was also financially supported by the National Natural Science Foundation Project (51575370) and 2014 Shanxi Province Coal Based Key Scientific and Technological Project (MJ2014-09).

ORCID iD

Xiaoxia Zhao

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