Investigation into the effect of common factors on rolling resistance of belt conveyor

Abstract

Since indentation rolling resistance accounts for the major part of total resistance of belt conveyor, it is important to compute it using a proper method during the design and application study of the belt conveyor. First, an approximate formula for computing the indentation rolling resistance is offered. In this formula, a one-dimensional Winkler foundation and a three-parameter viscoelastic Maxwell solid model of the belt backing material are used to determine the resistance to motion of a conveyor belt over idlers. With the help of this formula, the authors analyze the effect of common factors on the rolling resistance. Finally, experiments are carried out under certain condition compared with theoretical analysis. A reasonable correlation exists between the experimental results and the theoretical formulae.

Keywords

Viscoelastic property Maxwell solid model Winkler foundation rolling resistance experiment

Introduction

When belt backing passes over a series of idlers of belt conveyor, the inelastic deformation and indentation of backing material resulted from viscoelastic property can cause a type of running resistance that is defined as indentation rolling resistance. What is more, energy loss due to indentation rolling resistance is generally considered to be dominant loss mechanism. Vieweg et al.¹ elaborated the dynamic shear property of styrene–butadiene vulcanizates filled with carbon black. O’Shea et al.² proposed the effect of viscoelastic property measurements on the predicted rolling resistance. Fletcher and Gent,³ Payne,^4,5 Ulmer,⁶ Drozdov and Dorfmann,⁷ Vieweg et al.,^1,8 Wang,^9,10 Qiu,¹¹ and Qiu and Chai¹² presented the full two-dimensional model to determine the pressure distribution between the idler and belt backing. Hager et al.’s¹³ article developed the influence of the pulley cover compounds on resistance. Lodewijks¹⁴ investigated the connection between dynamic properties of rubber and energy loss due to rolling resistance. Wheeler¹⁵ analyzed the indentation rolling resistance with the help of finite element model (FEM) instead of full two-dimensional model. Lauhoff¹⁶ focused on the main factor, such as belt speed, that can affect the energy efficiency. Nordell¹⁷ discussed various types of running resistance on the belt conveyor. Jonkers¹⁸ carried out a lot of field experiments to deduce a formula to calculate indentation rolling resistance. Mao Jun^19,20 pointed out that the viscoelasticity of belt primarily causes the indentation resistance. Wheeler and Munzenberger²¹ presented the influence of various factors on the indentation rolling resistance using experimental data.

In this article, the authors discuss the viscoelastic property of the backing material and select Maxwell solid model to reflect the viscoelastic property first. Subsequently, with the help of Maxwell solid model in combination with the Winkler foundation deformation, a formula of indentation rolling resistance is inferred to analyze the effect of common factors on rolling resistance. Then, the authors propose an intelligent optimization algorithm to solve the formula. Finally, theoretical result when compared with the test result indicates that the formula has a reasonable reliability.

Theory of indentation rolling resistance

Three-parameter viscoelastic Maxwell model

Generally, the rubber compound which often acts as the backing material of belt often has the viscoelastic property. In order to reflect this property, various arrangements of springs and dashpots are used. Especially, Rudolphi and Reicks²² adopt 2N + 1 Maxwell model to express the viscoelastic characteristic. What is more, the superiority of using 2N + 1 Maxwell model is that it can achieve a higher accuracy for revealing the viscoelastic characteristic. It is worth noting that the theory of this article is based on the Rudolphi and Reicks’ article. However, in actual practice, the three-parameter Maxwell solid model is often used to reflect the viscoelastic property similar to the one presented in Lodewijks’s²³ article and adapted from May et al.²⁴ As for this model, it includes a single spring in parallel with another spring and a dashpot as shown in Figure 1, where the constants E ₁ and η ₁ represent the elastic and dissipative elements of the model and E ₀ is referred to the equilibrium modulus.

Figure 1.

Mechanical elements of three-parameter Maxwell model.

When the backing material selects the three-parameter Maxwell model to represent the viscoelastic property, the stress response function of the backing material is given by the Prony series²⁰

Ψ (t) = E_{0} + E_{1} \times \exp (\frac{- t}{τ})

(1)

where τ is the relaxation time, which is the characteristic period of the three-parameter Maxwell model

τ = \frac{η_{1}}{E_{1}}

Rubber properties and material characterization

Usually, a cyclic mode at various frequencies is utilized to achieve the viscoelastic property through Fourier analysis. As for a sinusoidal strain history of the form ε = ε ₀ sin(ωt), ω is the angular frequency and ε ₀ is the amplitude, after an initial strain transitory state, the stress of the backing material follows the strain in frequency and at a delayed phase or angle δ. In all, the connection between stress response and strain input is defined as follows

σ (ω) = E^{*} (ω) ε_{0} \sin (ω t + δ)

(2)

The stress is expressed by σ(ω) and the magnitude of the complex modulus is denoted by

E^{*} (ω) = \sqrt{E' {(ω)}^{2} + E ″ {(ω)}^{2}}

with real and imaginary components E′(ω) and E″(ω), called the storage and loss moduli, respectively. Furthermore, the storage and loss moduli of equation (2) are related to the mechanical element values by the Prony series

E' (ω) = E_{0} + E_{1} \times \frac{ω^{2} \times {(τ)}^{2}}{1 + {(ω τ)}^{2}}

(3)

E ″ (ω) = E_{1} \times \frac{{(ω τ)}^{2}}{1 + {(ω τ)}^{2}}

(4)

In this case, the parameter of δ deserves special attention because it has an important impact on the value of indentation rolling resistance by way of loss factor tan(δ). In this case, the loss factor tan(δ) is defined by

\tan (δ) = \frac{E ″}{E'} = \frac{ω η_{1} E_{1}^{2}}{E_{0} E_{1}^{2} + ω^{2} η_{1}^{2} (E_{1} + E_{0})}

In general, the mechanical properties, storage and loss moduli, are usually measured dynamically using a dynamic mechanical analyzer (DMA) and also by testing in a pure shear mode, and tests are performed in a cyclic mode. In this article, a specimen of the rubber material taken from the backing of a typical belt material is shown in Figure 2.

Figure 2.

Specimen of the rubber material.

Indentation resistance model

Due to the idlers are generally made of a relatively hard material, such as steel, compared with the much softer material of the backing material as shown in Figure 3, the belt will cause indentation on the contact region between idlers and belt. Furthermore, in virtue of viscoelastic property, when the belt is moving over the idlers, the contact area will cause an asymmetric indentation about the center of idler resulting in an asymmetric pressure distribution between idlers and belt. It produces the indentation rolling resistance of belt as shown in Figure 4.

Figure 3.

Stationary idler and belt.

Figure 4.

Geometric model of stationary idler and belt versus geometric model of motioning idler and belt.

From Figure 4, it can be seen that when idlers are in stationary state, the contact area is symmetric and contact length is 2a₀ ; however, once idler is in motion, the contact area is asymmetric and the absolute value of the first contact point “a” is greater than the absolute value of the point of departure b—/a/ > /−b/.

Generally, a convenient approach to determine the pressure distribution at any point of the contact area is to assume that the backing material can be modeled by one-dimensional Winkler foundation model (see Figure 5). The viscoelastic foundation of depth h rests on a rigid base and is compressed by the rigid idler. Because of the Winkler foundation model, it implies that shear between adjacent elements of the model is ignored. What is more, if the indentation depth h₁ is small compared to the thickness of the backing material h, the one-dimensional Winkler foundation model can be applied to approximate the deformation of the backing material due to the indentation of the idler.

Figure 5.

Winkler foundation model.

In order to discuss the indentation resistance, the authors first study the relationship between the stationary idler and belt. When idlers are in stationary state, the contact area is symmetric as shown in Figure 4.

Generally, compared to the idler radius R, the maximum indentation depth h₀ is very small, and h represents the thickness of the backing material. The contact length between belt and idler is 2a₀ and symmetric with Y-axis, when belt is in stationary state ( $h_{0} << R$ , $a_{0} << R$ , $a_{0}^{2} \approx 2 {Rh}_{0}$ ).

The indentation amount at a point X on the surface of the idler is an approximated parabola as follows

Δ Y = \sqrt{(R^{2} - X^{2})} - R + h_{0} \approx h_{0} - \frac{X^{2}}{2 R}

(5)

Hence, the compressional strain of the backing material at the coordinate X, modeled as a Winkler foundation through the backing, is as follows

\begin{matrix} ε_{0} & = \frac{h_{0}}{h} [1 - \frac{1}{2} {(\frac{\sqrt{2} X}{a_{0}})}^{2}] \\ \approx \frac{h_{0}}{h} \sin (\frac{π}{2} - β) = (\frac{h_{0}}{h}) \cos (β) \end{matrix}

(6)

β = \frac{\sqrt{2} X}{a_{0}} (- a_{0} < X < a_{0})

From equation (1), Ψ(0) = E ₁ + E ₂, the compressional stress between idler and belt is as follows

σ_{0} = \frac{h_{0}}{h} (E_{0} + E_{1}) \cos β

(7)

At equilibrium, the vertical load F_v and the resultant of the stress distribution must be in balance such that

\begin{matrix} F_{v} = \\ \int_{a_{0}}^{- a_{0}} σ_{0} \times dx = \frac{\sqrt{2} h_{0} \times a_{0}}{h} \times (E_{0} + E_{1}) \times \sin \sqrt{2} \\ \approx \frac{a_{0}^{3}}{R \times h \times \sqrt{2}} \times (E_{0} + E_{1}) \end{matrix}

(8)

There are following equations as shown

\sin (\sqrt{2}) \approx 1, a_{0} = {(\frac{π \times F_{v} \times R \times h}{2 (E_{0} + E_{1})})}^{1 / 3}

When the belt backing is passing over an idler at uniform speed v as shown in Figure 4 (in practice, the speed of belt often varies from 0.1 to 10 m/s), the contact length between belt and idler is asymmetric of the Y-axis resulting in asymmetric stress distribution. Finally, the asymmetric stress distribution causes the indentation rolling resistance in the horizontal direction.

Generally, the contact length (a + b), both the point of first contact “a” and the point of departure “b,” is determined by the load between belt and idler. What is more, for a given belt speed v, the relationship between belt speed v and angular frequency ω is taken to be as follows

ω = \frac{π \times ν}{(a + b)}

When the belt is in motion, the compressional strain of the backing material at the coordinate X is as follows and h₁ is the maximum indentation depth of the backing material, when belt is in motion

ε = \frac{h_{1}}{h} [1 - \frac{1}{2} {(\frac{\sqrt{2}}{a} X)}^{2}] \approx \frac{h_{1}}{h} \sin (\frac{π}{2} - β) = \frac{h_{1}}{h} \cos β

(9)

β = \frac{\sqrt{2} X}{a}, - b < x < a, a^{2} \approx 2 {Rh}_{1}

At a constant speed, the deformation process of backing is in the essentially steady state with respect to the Eulerian coordinate X = a − vt, where t denotes the time and a the point of contact of the belt and idler (X is fixed with respect to the idler but not rotating with it). Using the translating coordinate speed relationship which is shown as follows

X = a - ν t, \frac{dx}{dt} = - ν

the compressional strain of an arbitrary point on the backing material (equation (9)) may be expressed as a function of t as follows

ε (t) = \frac{h_{1}}{h} \sin (\frac{\sqrt{2}}{a} ν t)

(10)

As for a linear viscoelastic material and a one-dimensional state of stress, the stress response function for a prescribed strain history is as follows

σ (t) = \int_{0}^{t} Ψ (t - τ) ε (t) d τ

(11)

Applying the Winkler foundation and the three-parameter Maxwell model, with the help of equations (1), (9), and (10) and the following equations

\frac{π}{2} \approx \sqrt{2}, a^{2} \approx 2 {Rh}_{1}

The compressional stress for an arbitrary fixed point in the contact area may be expressed as a function of t as follows

\begin{matrix} σ_{y} (t) = & a^{2} {\frac{E_{0}}{2 Rh} \times \frac{t ν}{a} \times \frac{2 a - t ν}{a} + \frac{E_{1} \times k}{Rh} \times (1 + k) \\ \times (1 - \exp (\frac{- 1}{k} \times \frac{t ν}{a})) - \frac{E_{1} \times k}{Rh} \times \frac{t ν}{a}} \end{matrix}

(12)

Using the transformation t = (a − X)/v, the compressional stress of equation (11) can be expressed as a function of X as follows that yields the stress distribution between idler and belt backing

\begin{matrix} σ_{y} (X) & = a^{2} \times {\frac{E_{0}}{2 Rh} \times \frac{t ν}{a} \times \frac{2 a - t ν}{a} + \frac{E_{1} \times k}{Rh} \times (1 + k) \times (1 - \exp (\frac{- 1}{k} \times \frac{t ν}{a})) - \frac{E_{1} \times k}{Rh} \times \frac{t ν}{a}} \\ \approx \frac{a^{2} \times E_{1}}{2 Rh (1 + E_{1}^{2} \times a^{2} / 2 ν^{2} η_{1}^{2})} [\cos (\frac{π X}{2 a}) + \frac{2 {aE}_{1}}{π η_{1}} \sin (\frac{π X}{2 a})] - \frac{a^{2} \times E_{1}}{2 Rh (1 + E_{1}^{2} \times a^{2} / 2 ν^{2} η_{1}^{2})} \times (\frac{2 {aE}_{1}}{π η_{1} ν} \times \exp (\frac{E_{1} (X - a)}{ν η_{1}})) + \frac{a^{2} \times E_{0}}{2 Rh} \cos (\frac{π X}{2 a}) \end{matrix}

(13)

k = \frac{ν τ}{a}

Now, introducing the nondimensional length ζ = b/a, from Figure 4, it is not been ignored that the stress of equation (13) must be equated to zero at X = −b as follows

\begin{matrix} a^{2} & \times {\frac{E_{0}}{2 Rh} (1 + ς) (1 - ς) + \frac{E_{1} \times k}{R \times h} \\ \times (1 + k) (1 - \exp (\frac{1 + ς}{- K})) - \frac{E_{1} k}{Rh} (1 + ς)} = 0 \end{matrix}

(14)

When the belt moves at a constant speed v, the vertical load F_v is constant for a stationary moving belt leading to the fact that the load F_v and the resultant of the stress distribution σ(X) must also be in balance. Therefore

\begin{matrix} F_{v} & = \int_{- b}^{a} σ_{Y} (X) dX = \frac{E_{0} {(a + b)}^{2} (2 a - b)}{6 Rh} \\ + \frac{a^{2} E_{1} k}{Rh} (1 + k) (a + b - k \times a + k \times a \times \exp (\frac{- b - a}{ka}))_\frac{a \times E_{1} \times k}{R \times h} \times \frac{{(a + b)}^{2}}{2} \\ = \frac{E_{0} \times a^{3} \times {(1 + ς)}^{2} (2 - ς)}{6 Rh} + \frac{a^{3} \times E_{1} \times k}{Rh} (1 + k) (1 + ς - k + k \times \exp (\frac{1 + ς}{- k})) - \frac{a^{3} E_{1} k {(1 + ς)}^{2}}{2 Rh} \end{matrix}

(15)

Then, for given material parameters E₀, E₁ , and η₁ and belt system parameters F_v, R, h, B, and v, in order to acquire the value of indentation rolling resistance, the values of a and b which must satisfy equations (14) and (15) with the load F_v should be computed first. The simultaneous solution is readily accomplished by intelligent optimization algorithm as shown in Figure 6.

Figure 6.

Intelligent optimization algorithm.

Following this intelligent optimization algorithm to determine an accurate value of ζ = b/a, a, and b, the moment M of the stress distribution about the center of the idler is as follows

M = \int_{- b}^{a} X σ_{Y} (X) dX

Then the indentation rolling resistance F_h is as follows

\begin{matrix} F_{h} & = \frac{M}{R} = \frac{E_{0} a^{4}}{8 R^{2} h} {[{(\frac{b}{a})}^{2} - 1]}^{2} + \frac{a^{4} {kE}_{1}}{R^{2} h} \\ \times [k^{3} - \frac{k}{2} (1 + {(\frac{b}{a})}^{2}) + \frac{1}{3} (1 + {(\frac{b}{a})}^{3}) - k (1 + k) (k + \frac{b}{a}) \times \exp (\frac{- 1}{k} (\frac{a + b}{a}))] \\ = \frac{E_{0} a^{4}}{8 R^{2} h} {(ς^{2} - 1)}^{2} + \frac{a^{4} {kE}_{1}}{R^{2} h} [\frac{1 + ς^{3}}{3} + k^{3} - \frac{k (1 + ς^{3})}{3} - k (1 + k) (k + ς) \exp (\frac{1 + ς}{- k})] \end{matrix}

(16)

where R is the idler radius; (a + b) is the contact length; E ₀, E ₁, and η ₁ are the viscoelastic parameters of the backing material; and h is the thickness of the backing material, all have an effect on the indentation rolling resistance. As for the test belt which is the fabric belt, the viscoelastic parameters²⁰ are shown as follows: E ₀ = 7 × 106 Pa, E ₁ = 2.5 × 108 Pa, and η ₁ = 1875 N s/m² (Table 1).

Table 1.

Basic parameter of the test piece.

Test piece	Belt type	Width (mm)	Thickness of rubber material	Length (mm)
Fabric belt	400 × 4	500	6 (top)+1.5 (below)	5620

Analysis for theoretical results

As mentioned above, there is a close relationship between rolling resistance and belt system parameters F_v , R, h, B, and v. In this article, the authors mainly concentrate on the influence between the rolling resistance and load F_v and belt speed v, thickness of the backing material h, and idler radius R.

In general, the speed of belt varies from 0.1 to 10 m/s; meanwhile, the variation range of vertical load F_v is from 0 to 2500 N/m. Therefore, the authors apply some representative belt speed points to the test belt (1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 m/s) under some fixed vertical loads F_v (500, 1000, 1500, 2000, and 2500 N/m), idler diameters (89, 108, and 133 mm), thicknesses of backing rubber (2, 4, and 6 mm).

Tables 2 –28 show the contact length a + b under different conditions; what is more, these values lay a foundation of the following research in this article.

Table 2.

Contact length a + b (mm) versus R = 44.5 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	11	8.3	6.9	6.0	5.3	4.9	4.5	4.2	4.0	3.8
1000	16	12	9.7	8.4	7.5	6.9	6.4	6.0	5.6	5.3
1500	19	14	12	10	9.3	8.7	7.8	7.3	6.9	5.5
2000	23	17	14	12	11	9.9	9.3	8.4	8.0	7.5
2500	25	19	15	13	12	11	10	9.4	8.9	8.4

Table 3.

Contact length a + b (mm) versus R = 54 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	13	9.1	7.5	6.6	5.9	5.4	5.0	4.6	4.4	4.2
1000	19	13	11	9.3	8.3	7.6	7.0	6.6	6.2	5.9
1500	23	16	13	11	10	9.3	8.6	8.0	7.6	7.2
2000	26	19	15	13	12	11	9.9	9.3	8.8	8.3
2500	29	20	17	15	13	12	11	10	9.8	9.3

Table 4.

Contact length a + b (mm) versus R = 66.5 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	15	10	8.4	7.3	6.5	6.0	5.5	5.2	4.8	4.6
1000	21	15	12	10	9.2	8.4	7.8	7.3	6.9	6.5
1500	25	17	15	13	11	10	9.5	8.9	8.4	8.0
2000	29	21	17	15	13	12	11	10	9.7	9.2
2500	32	23	18	16	15	13	12	11.5	10.8	10.3

Table 5.

Contact length a + b (mm) versus R = 44.5 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	9.7	6.9	5.6	4.9	4.4	4.0	3.7	3.4	3.2	3.1
1000	14	9.7	8.0	6.9	6.2	5.6	5.2	4.9	4.6	4.4
1500	17	12	9.7	8.4	7.5	6.9	6.4	6.0	5.6	5.3
2000	19	13	11	9.7	8.7	8.0	7.4	6.9	6.5	6.2
2500	22	15	13	11	9.7	8.9	8.2	7.7	7.3	6.9

Table 6.

Contact length a + b (mm) versus R = 54 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	11	7.6	6.1	5.4	4.8	4.4	4.1	3.8	3.6	3.4
1000	15	11	8.7	7.6	6.8	6.2	5.7	5.4	5.1	4.8
1500	19	13	11	9.3	8.3	7.6	7.1	6.6	6.2	5.9
2000	21	15	12	11	9.6	8.8	8.1	7.5	7.2	6.8
2500	24	17	14	12	11	9.8	9.1	8.5	8.0	7.6

Table 7.

Contact length a + b (mm) versus R = 66.5 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	12	8.4	6.9	6.0	5.3	4.9	4.5	4.2	4.0	3.8
1000	17	12	9.7	8.4	7.5	6.9	6.4	6.0	5.6	5.3
1500	21	15	12	10	9.2	8.4	7.8	7.1	6.9	6.5
2000	24	17	14	12	11	9.7	9.0	8.4	7.9	7.6
2500	27	18	15	13	12	11	10	9.4	8.9	8.4

Table 8.

Contact length a + b (mm) versus R = 44.5 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	6.9	4.9	4.0	3.4	3.1	3.0	2.6	2.4	2.3	2.1
1000	9.7	6.9	5.6	4.9	4.4	4.0	3.7	3.4	3.2	3.1
1500	12	8.4	6.9	6.0	5.3	4.9	4.5	4.2	4.0	3.8
2000	14	9.7	8.0	6.9	6.2	5.6	5.2	4.9	4.6	4.4
2500	15	11	8.9	7.7	6.9	6.3	5.8	5.4	5.1	4.9

Table 9.

Contact length a + b (mm) versus R = 54 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	7.6	5.4	4.4	3.8	3.4	3.1	2.9	2.7	2.5	2.4
1000	11	7.6	6.2	5.4	4.8	4.5	4.1	3.8	3.6	3.4
1500	13	9.3	7.6	6.6	5.9	5.4	5.0	4.6	4.4	4.2
2000	15	11	8.8	7.6	6.8	6.2	5.7	5.4	5.1	4.8
2500	17	12	9.8	8.5	7.6	6.9	6.4	6.0	5.7	5.4

Table 10.

Contact length a + b (mm) versus R = 66.5 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	8.4	6.0	4.9	4.2	3.8	3.4	3.2	3.0	2.8	2.7
1000	12	8.4	6.9	6.0	5.3	4.9	4.5	4.2	4.0	3.8
1500	15	10	8.4	7.3	6.5	6.0	5.5	5.2	4.9	4.6
2000	17	12	9.7	8.4	7.5	6.9	6.4	6.0	5.6	5.3
2500	19	13	11	9.4	8.4	7.7	7.1	6.7	6.3	6.0

Table 11.

Contact length a (mm) versus R = 44.5 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	5.8	4.4	3.6	3.1	2.8	2.6	2.4	2.2	2.1	2.0
1000	8.4	6.3	5.1	4.4	3.9	3.6	3.4	3.1	2.9	2.8
1500	10	7.3	6.3	5.2	4.9	4.6	4.1	3.8	3.6	2.9
2000	12	8.9	7.3	6.3	5.8	5.2	4.9	4.4	4.2	3.9
2500	13	10	7.9	6.8	6.3	5.8	5.2	4.9	4.7	4.4

Table 12.

Contact length b (mm) versus R = 44.5 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	5.2	3.9	3.3	2.9	2.5	2.3	2.1	2.0	1.9	1.8
1000	7.6	5.7	4.6	4.0	3.6	3.3	3.0	2.9	2.7	2.5
1500	9	6.7	5.7	4.8	4.4	4.1	3.7	3.5	3.3	2.6
2000	11	8.1	6.7	5.7	5.2	4.7	4.4	4.0	3.8	3.6
2500	12	9	7.1	6.2	5.7	5.2	4.8	4.5	4.2	4.0

Table 13.

Contact length a (mm) versus R = 54 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	6.8	4.8	3.9	3.5	3.1	2.8	2.6	2.4	2.3	2.2
1000	10	6.8	5.8	4.9	4.4	4.0	3.7	3.5	3.3	3.1
1500	12	8.4	6.8	5.8	5.2	4.9	4.5	4.2	4.0	3.8
2000	14	10	7.9	6.8	6.3	5.8	5.2	4.9	4.6	4.4
2500	15.2	10.4	9	7.9	6.8	6.3	5.8	5.2	5.1	4.9

Table 14.

Contact length b (mm) versus R = 54 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	6.2	4.3	3.6	3.1	2.9	2.6	2.4	2.2	2.1	2.0
1000	9	6.2	5.2	4.4	3.9	3.6	3.3	3.1	2.9	2.8
1500	11	7.6	6.2	5.2	4.8	4.4	4.1	3.8	3.6	3.4
2000	12	9	7.1	6.2	5.7	5.2	4.7	4.4	4.2	3.9
2500	13.8	9.6	8	7.1	6.2	5.7	5.2	4.8	4.7	4.4

Table 15.

Contact length a (mm) versus R = 66.5 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	7.9	5.2	4.4	3.8	3.4	3.1	2.9	2.7	2.5	2.4
1000	11	7.9	6.3	5.2	4.8	4.4	4.1	3.8	3.6	3.4
1500	13	8.9	7.9	6.8	5.8	5.2	5.0	4.7	4.4	4.2
2000	15	11	8.9	7.9	6.8	6.3	5.8	5.2	5.1	4.8
2500	17	12	9.4	8.4	7.9	6.8	6.3	6.0	5.7	5.4

Table 16.

Contact length b (mm) versus R = 66.5 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	7.1	4.8	4.0	3.5	3.1	2.9	2.6	2.5	2.3	2.2
1000	10	7.1	5.7	4.8	4.4	4.0	3.7	3.5	3.3	3.1
1500	12	8.1	7.1	6.2	5.2	4.8	4.5	4.2	4.0	3.8
2000	14	10	8.1	7.1	6.2	5.7	5.2	4.8	4.6	4.4
2500	15	11	8.6	7.6	7.1	6.2	5.7	5.5	5.1	4.9

Table 17.

Contact length a (mm) versus R = 44.5 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	5.1	3.6	2.9	2.6	2.3	2.1	1.9	1.8	1.7	1.62
1000	7.3	5.1	4.2	3.6	3.3	2.9	2.7	2.6	2.4	2.3
1500	8.9	6.3	5.1	4.4	3.9	3.6	3.4	3.1	2.9	2.8
2000	10	6.8	5.8	5.1	4.6	4.2	3.9	3.6	3.4	3.3
2500	12	7.9	6.8	5.8	5.1	4.7	4.3	4.0	3.8	3.6

Table 18.

Contact length b (mm) versus R = 44.5 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	4.6	3.3	2.7	2.3	2.1	1.9	1.8	1.6	1.5	1.48
1000	6.7	4.6	3.8	3.3	2.9	2.7	2.5	2.3	2.2	2.1
1500	8.1	5.7	4.6	4.0	3.6	3.3	3.0	2.9	2.7	2.5
2000	9	6.2	5.2	4.6	4.1	3.8	3.5	3.3	3.1	2.9
2500	10	7.1	6.2	5.2	4.6	4.2	3.9	3.7	3.5	3.3

Table 19.

Contact length a (mm) versus R = 54 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	5.8	4.0	3.2	2.8	2.5	2.3	2.2	2.0	1.9	1.8
1000	7.9	5.8	4.6	4.0	3.6	3.3	3.0	2.8	2.7	2.5
1500	10	6.8	5.8	4.9	4.4	4.0	3.7	3.5	3.3	3.1
2000	11	7.9	6.3	5.8	5.0	4.6	4.3	3.9	3.8	3.6
2500	13	9	7.3	6.3	5.8	5.1	4.8	4.5	4.2	4.0

Table 20.

Contact length b (mm) versus R = 54 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	5.2	3.6	2.9	2.6	2.3	2.1	1.9	1.8	1.7	1.6
1000	7.1	5.2	4.1	3.6	3.2	2.9	2.7	2.6	2.4	2.3
1500	9	6.2	5.2	4.4	3.9	3.6	3.4	3.1	2.9	2.8
2000	10	7.1	5.7	5.2	4.6	4.2	3.8	3.6	3.4	3.2
2500	11	8	6.7	5.7	5.2	4.7	4.3	4.0	3.8	3.6

Table 21.

Contact length a (mm) versus R = 66.5 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	6.3	4.4	3.6	3.1	2.8	2.6	2.4	2.2	2.1	2.0
1000	8.9	6.3	5.1	4.4	3.9	3.6	3.4	3.1	2.9	2.8
1500	11	7.9	6.3	5.3	4.8	4.4	4.1	3.7	3.6	3.4
2000	13	8.9	7.3	6.3	5.8	5.1	4.7	4.4	4.1	4.0
2500	14	9.4	7.9	6.8	6.3	5.8	5.2	4.9	4.7	4.4

Table 22.

Contact length b (mm) versus R = 66.5 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	5.7	4.0	3.3	2.9	2.5	2.3	2.1	2.0	1.9	1.8
1000	8.1	5.7	4.6	4.0	3.6	3.3	3.0	2.9	2.7	2.5
1500	10	7.1	5.7	4.7	4.4	4.0	3.7	3.4	3.3	3.1
2000	11	8.1	6.7	5.7	5.2	4.6	4.3	4.0	3.8	3.6
2500	13	8.6	7.1	6.2	5.7	5.2	4.8	4.5	4.2	4.0

Table 23.

Contact length a (mm) versus R = 44.5 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	3.6	2.6	2.1	1.8	1.6	1.57	1.4	1.25	1.2	1.1
1000	5.1	3.6	2.9	2.6	2.3	2.1	1.9	1.8	1.7	1.62
1500	6.3	4.4	3.6	3.1	2.8	2.6	2.4	2.2	2.1	2.0
2000	7.3	5.1	4.2	3.6	3.3	2.9	2.7	2.6	2.4	2.3
2500	7.9	5.8	4.7	4.0	3.6	3.3	3.0	2.8	2.7	2.6

Table 24.

Contact length b (mm) versus R = 44.5 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	3.3	2.3	1.9	1.6	1.5	1.43	1.2	1.15	1.1	1.0
1000	4.6	3.3	2.7	2.3	2.1	1.9	1.8	1.6	1.5	1.48
1500	5.7	4.0	3.3	2.9	2.5	2.3	2.1	2.0	1.9	1.8
2000	6.7	4.6	3.8	3.3	2.9	2.7	2.5	2.3	2.2	2.1
2500	7.1	5.2	4.2	3.7	3.3	3.0	2.8	2.6	2.4	2.3

Table 25.

Contact length a (mm) versus R = 54 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	4.0	2.8	2.3	2.0	1.8	1.6	1.5	1.4	1.3	1.25
1000	5.8	4.0	3.3	2.8	2.5	2.4	2.2	2.0	1.9	1.8
1500	6.8	4.9	4.0	3.5	3.1	2.8	2.6	2.4	2.3	2.2
2000	7.9	5.8	4.6	4.0	3.6	3.3	3.0	2.8	2.7	2.5
2500	9.0	6.3	5.1	4.5	4.0	3.6	3.4	3.1	3.0	2.8

Table 26.

Contact length b (mm) versus R = 54 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	3.6	2.6	2.1	1.8	1.6	1.5	1.4	1.3	1.2	1.15
1000	5.2	3.6	2.9	2.6	2.3	2.1	1.9	1.8	1.7	1.6
1500	6.2	4.4	3.6	3.1	2.8	2.6	2.4	2.2	2.1	2.0
2000	7.1	5.2	4.2	3.6	3.2	2.9	2.7	2.6	2.4	2.3
2500	8	5.7	4.7	4.0	3.6	3.3	3.0	2.9	2.7	2.6

Table 27.

Contact length a (mm) versus R = 66.5 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	4.4	3.1	2.6	2.2	2.0	1.8	1.7	1.6	1.46	1.4
1000	6.3	4.4	3.6	3.1	2.8	2.6	2.4	2.2	2.1	2.0
1500	7.9	5.2	4.4	3.8	3.4	3.1	2.9	2.7	2.6	2.4
2000	8.9	6.3	5.1	4.4	3.9	3.6	3.4	3.1	2.9	2.8
2500	10	6.8	5.8	4.9	4.4	4.0	3.7	3.5	3.3	3.1

Table 28.

Contact length b (mm) versus R = 66.5 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	4.0	2.9	2.3	2.0	1.8	1.6	1.5	1.4	1.34	1.3
1000	5.7	4.0	3.3	2.9	2.5	2.3	2.1	2.0	1.9	1.8
1500	7.1	4.8	4.0	3.5	3.1	2.9	2.6	2.5	2.3	2.2
2000	8.1	5.7	4.6	4.0	3.6	3.3	3.0	2.9	2.7	2.5
2500	9	6.2	5.2	4.5	4.0	3.7	3.4	3.2	3.0	2.9

It can be seen from Figures 7 –10, the contact length a + b decreases nonlinearly with increasing belt speed v. However, there is a phenomenon worthy of attention that while the value of belt speed is in the lower level, the contact length decreases rapidly with increasing belt speed. As for this style of test belt, the threshold of belt speed v is 6 m/s. Furthermore, from Figures 7, 9, and 10, it can be inferred that once the value of viscoelastic property of the backing material is established, the threshold of belt speed is not correlated with idler radius R and thickness of the backing material h. It means that when we consider the relationship between indentation rolling resistance, by way of contact length a + b and speed v, we should not ignore the effect of viscoelastic property.

Figure 7.

Contact length a + b (mm) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 44.5 mm and h = 6 mm.

Figure 8.

Contact length a + b (mm) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 54 mm and h = 6 mm.

Figure 9.

Contact length a + b (mm) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 66.5 mm and h = 6 mm.

Figure 10.

Contact length a + b (mm) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 54 mm and h = 4 mm.

From Figures 11 –13, the contact length a + b increases linearly (a + b = k·F_v + c) with increasing vertical load F_v. What is more, the linear regular is applicable to a variety of circumstances which is not correlated with idler radius R, thickness of the backing material h, and belt speed v.

Figure 11.

Contact length a + b (mm) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 66.5 mm and h = 2 mm.

Figure 12.

Contact length a + b (mm) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 44.5 mm and h = 6 mm.

Figure 13.

Contact length a + b (mm) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 54 mm and h = 4 mm.

From Tables 29 –34, it can be seen that the value of slope k and increment c decreases with increasing speed v. It means that the effect of vertical load F_v on the indentation rolling resistance will be weakened with increasing speed v (Tables 35 –43). It can also be inferred that when we investigated the influence of vertical load F_v on rolling resistance, the belt speed also played a role in it.

Table 29.

Slope k versus vertical load F_v , R = 44.5 mm, and h = 6 mm under different v.

v (m/s)	2	4	6	8	10
k	5.3e−3	3.5e−3	3.0e−3	2.6e−3	2.2e−3

Table 30.

Increment c versus vertical load F_v , R = 44.5 mm, and h = 6 mm under different v.

v (m/s)	2	4	6	8	10
c	6.1	4.6	3.7	3.2	2.8

Table 31.

Slope k versus vertical load F_v , R = 54 mm, and h = 4 mm under different v.

v (m/s)	2	4	6	8	10
k	4.6e−3	3.3e−3	2.7e−3	2.3e−3	2.1e−3

Table 32.

Increment c versus vertical load F_v , R = 54 mm, and h = 4 mm under different v.

v (m/s)	2	4	6	8	10
c	5.9	4.1	3.3	2.9	2.6

Table 33.

Slope k versus vertical load F_v , R = 66.5 mm, and h = 2 mm under different v.

v (m/s)	2	4	6	8	10
k	3.5e−3	2.6e−3	2.1e−3	1.8e−3	1.6e−3

Table 34.

Increment c versus vertical load F_v , R = 66.5 mm, and h = 2 mm under different v.

v (m/s)	2	4	6	8	10
c	4.6	3.2	2.6	2.3	2.3

Table 35.

Indentation rolling resistance F_h (N/m) versus R = 44.5 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	21	16	13	11	10	9.2	8.3	7.6	7.3	6.9
1000	66	49	38	32	28	26	24	22	20	19
1500	117	78	72	55	54	53	44	40	38	21
2000	200	143	113	95	91	78	75	62	60	54
2500	252	208	144	120	117	108	92	87	84	77

Table 36.

Indentation rolling resistance F_h (N/m) versus R = 54 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	23	12	12	11	9.6	8.3	7.7	6.8	6.6	6.3
1000	80	42	38	30	26	24	22	21	19	18
1500	136	82	62	50	45	44	40	37	35	33
2000	295	141	98	81	80	73	61	58	54	51
2500	302	156	147	128	101	95	85	72	75	72

Table 37.

Indentation rolling resistance F_h (N/m) versus R = 66.5 mm and h = 6 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	25	12	11	9.3	8.1	7.5	6.8	6.5	5.7	5.5
1000	69	45	33	24	23	21	20	18	17	16
1500	113	64	65	54	41	36	36	34	31	30
2000	172	124	93	84	67	63	56	47	49	45
2500	330	161	110	102	104	79	73	72	68	64

Table 38.

Indentation rolling resistance F_h (N/m) versus R = 44.5 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	21	13	10	9.5	8.3	7.4	6.7	6.0	5.6	5.5
1000	62	39	31	26	24	21	19	18	17	16
1500	118	74	57	48	42	39	36	33	30	29
2000	176	94	84	75	67	61	56	51	48	47
2500	429	150	137	110	92	85	76	71	68	64

Table 39.

Indentation rolling resistance F_h (N/m) versus R = 54 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	21	12	9.3	8.4	7.3	6.7	6.4	5.7	5.4	5.0
1000	56	39	28	24	21	19	17	16	15	14
1500	120	64	57	45	39	36	33	31	29	27
2000	158	102	73	75	60	55	51	45	45	42
2500	362	154	116	96	92	76	72	66	61	58

Table 40.

Indentation rolling resistance F_h (N/m) versus R = 66.5 mm and h = 4 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	18	11	8.9	7.6	6.6	6.2	5.6	5.1	4.9	4.6
1000	53	33	25	21	19	17	16	15	14	13
1500	104	67	48	37	35	32	29	25	26	24
2000	239	96	76	64	61	49	45	42	39	38
2500	422	122	99	80	76	69	57	54	52	47

Table 41.

Indentation rolling resistance F_h (N/m) versus R = 44.5 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	14	9.8	7.6	6.2	5.7	6.2	4.7	4.1	4.0	3.3
1000	41	27	21	19	17	15	13	12	11	11
1500	80	49	40	34	30	28	25	23	22	21
2000	124	77	63	52	48	41	38	37	33	32
2500	166	115	88	73	65	59	53	49	46	45

Table 42.

Indentation rolling resistance F_h (N/m) versus R = 54 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	13	8.6	6.9	5.9	5.2	4.6	4.4	4.0	3.5	3.4
1000	42	25	20	17	15	15	13	11	11	10
1500	68	47	37	32	28	25	23	20	20	19
2000	113	78	57	48	43	39	34	33	31	29
2500	176	101	78	69	60	53	50	45	44	41

Table 43.

Indentation rolling resistance F_h (N/m) versus R = 66.5 mm and h = 2 mm under different F_v .

F_v (N/m)	v (m/s)
F_v (N/m)	1	2	3	4	5	6	7	8	9	10
500	12	7.7	6.4	5.2	4.8	4.1	3.9	3.7	3.3	3.2
1000	36	22	18	15	13	12	11	10	9.8	9.3
1500	74	37	32	28	24	23	20	19	18	17
2000	106	66	51	43	37	35	33	30	27	26
2500	158	84	75	60	53	48	44	42	39	37

From Figure 14, it can be seen that the contact length a + b and peak value of stress are inversely proportional to the belt speed except for belt speed v = 10 m/s. What is more, it can be inferred that the pressure profile between idler and belt is comparatively symmetric about the centerline of X = 0 at a belt speed of 2 m/s. However, as the belt speed increases, the stress profile becomes more asymmetric. It is noticeable that a “hump” of the stress forms behind the centerline X = 0.

Figure 14.

Contact length a + b (mm) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 66.5 mm and h = 2 mm.

From Figures 15 –17, it can be inferred that the contact length (a + b) and the peak value of stress are proportional to the vertical load F_v . Furthermore, the greater the value of vertical load, the more asymmetric the stress profile.

Figure 15.

Calculated pressure distribution for 44.5 mm under a simulated vertical load of 500 N/m at a belt speed of 2, 4, 6, 8, and 10 m/s.

Figure 16.

Calculated pressure distribution for 44.5 mm under a simulated belt speed of 2 m/s at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m.

Figure 17.

Calculated pressure distribution for 44.5 mm under a simulated belt speed of 6 m/s at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m.

From Figures 18 –26, it can be seen that the influence of belt speed v on the indentation rolling resistance F_h is complex. Overall, when the belt speed is less than the threshold (6 m/s), the rolling resistance decreases obviously with an increasing belt speed. Because the indentation rolling resistance is the main resistance, when belt conveyor is in motion, it means that increasing the belt speed is beneficial to reduce indentation rolling resistance resulting in cutting down the energy loss of belt. However, a noteworthy phenomenon is that when the belt speed is above the threshold, the value of indentation rolling resistance plateaus even through the belt speed still increases.

Figure 18.

Indentation rolling resistance (N/m) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 44.5 mm and h = 6 mm.

Figure 19.

Indentation rolling resistance (N/m) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 54 mm and h = 4 mm.

Figure 20.

Indentation rolling resistance (N/m) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 66.5 mm and h = 2 mm.

Figure 21.

Indentation rolling resistance (N/m) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 54 mm and h = 6 mm.

Figure 22.

Indentation rolling resistance (N/m) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 66.5 mm and h = 4 mm.

Figure 23.

Indentation rolling resistance (N/m) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 44.5 mm and h = 2 mm.

Figure 24.

Indentation rolling resistance (N/m) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 66.5 mm and h = 6 mm.

Figure 25.

Indentation rolling resistance (N/m) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 44.5 mm and h = 4 mm.

Figure 26.

Indentation rolling resistance (N/m) versus belt speed v at a vertical load of 500, 1000, 1500, 2000, and 2500 N/m under constant values of R = 54 mm and h = 2 mm.

From Figures 18 –20, it can be seen that once the value of viscoelastic property of the backing material is established, the threshold of belt speed is not correlated with idler radius R and thickness of the backing material h. This phenomenon infers that when we consider the relationship between indentation rolling resistance and belt speed, we should not ignore the effect of viscoelastic property.

What is more, from Figures 18 –26, it can be concluded that the effect of belt speed on rolling resistance correlated with the vertical load F_v , when the vertical load was at a lower level, such as 500 N/m, and the effect is very distinct that when belt speed increases from 1 to 2 m/s, the rolling resistance decreases obviously. However, once the vertical load is above the threshold (2000 N/m), even if the belt speed increases from 1 to 10 m/s, the rolling resistance remains unaltered.

From Figures 27 –35, it can be seen that the rolling resistance F_h increases linearly with increasing vertical load F_v . What is more, the growth rate of rolling resistance is relevant with the belt speed v. Finally, a conclusion illustrates the filiation between rolling resistance and belt speed, vertical load is given by way of interpolant of MATLAB when idler radius R is 66.5 mm and thickness of the backing material h is 6 mm.

Figure 27.

Indentation rolling resistance (N/m) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 44.5 mm and h = 6 mm.

Figure 28.

Indentation rolling resistance (N/m) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 54 mm and h = 4 mm.

Figure 29.

Indentation rolling resistance (N/m) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 66.5 mm and h = 2 mm.

Figure 30.

Indentation rolling resistance (N/m) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 54 mm and h = 6 mm.

Figure 31.

Indentation rolling resistance (N/m) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 66.5 mm and h = 4 mm.

Figure 32.

Indentation rolling resistance (N/m) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 44.5 mm and h = 2 mm.

Figure 33.

Indentation rolling resistance (N/m) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 66.5 mm and h = 6 mm.

Figure 34.

Indentation rolling resistance (N/m) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 44.5 mm and h = 4 mm.

Figure 35.

Indentation rolling resistance (N/m) versus vertical load F_v at a belt speed of 2, 4, 6, 8, and 10 m/s under constant values of R = 54 mm and h = 2 mm.

From Figures 36 –38, it can be seen that the maximum rolling resistance will appear when the vertical load is maximum and the belt speed is minimum. However, in the variation range of vertical load and belt speed, the value of rolling resistance exists as an extreme value.

Figure 36.

Indentation rolling resistance (N/m) versus vertical load F_v and belt speed v under constant values of R = 66.5 mm and h = 6 mm.

Figure 37.

Indentation rolling resistance (N/m) versus vertical load F_v and belt speed v under constant values of R = 54 mm and h = 6 mm.

Figure 38.

Indentation rolling resistance (N/m) versus vertical load F_v and belt speed v under constant values of R = 44.5 mm and h = 6 mm.

Experiments and verification

In order to verify the accuracy of the theory, the authors need to design a reliable experimental apparatus. To get reliable test data, the apparatus is built to simulate a real belt conveyor.

Apparatus and test method

The experimental apparatus is composed of two components, namely, mechanical structure and data acquisition (DAQ) with signal processing. Finally, the function structure of indentation rolling resistance measure system is shown in Figure 39.

Figure 39.

Function structure of the indentation rolling resistance measure system.

On one hand, as for the mechanical structure (Figure 40), it mainly consists of drive pulley, frequency converter, tension pulley, test belt, carrying flat, test idler, and loading structure.

Figure 40.

Mechanical structure of experimental apparatus.

On the other hand, the DAQ with signal processing includes tension sensor, DAQ card, signal processing device, and a test program of PC with the help of LabVIEW as shown in Figure 41. It is noticeable that the style of tension sensor is S. As for the DAQ card, it plays a role in DAQ. The function of signal processing device is signal amplification.

Figure 41.

Data acquisition and signal processing.

As for the test program, it plays a part in DAQ, setting sampling frequency, signal filtering, and storage of signal as shown in Figure 42.

Figure 42.

Diagram of program of data acquisition.

Verification

Aimed at this experimental apparatus, the parameters which can be changed are belt speed v, load F_v , and idler radius R. In this experiment, the idler radius R is 44.5, 54, and 66.5 mm, as shown in Figure 43, and thickness of rubber h is 6 mm. Using the frequency converter, the belt speed can be changed. With the help of loading structure, different loads F_v can be applied on the test belt.

Figure 43.

Test idler.

As for the apparatus, because the maximum setting value of speed v is 6 m/s, in order to verify the theoretical results, the authors set the belt speed v at 2, 4, and 6 m/s and the load F_v at 500, 1000, and 1500 N/m, respectively.

In order to improve the reliability of test results, the variation of the belt speed v must satisfy the following conditions: first, test data must be measured in accordance with the increasing trend of belt speed v from 2 to 4 m/s at a constant vertical load F_v and recorded. Then, test data should also be measured in accordance with the decreasing trend of belt speed v from 2 to 4 m/s at a constant vertical load F_v and recorded. Finally, experiment must be retested once time and average value of four test results act as the ultimate value, and the standard deviation of ultimate value must be calculated.

Generally, it can be seen from Figures 44 –46 that the test result is consistent with the theoretical result. However, when the belt speed v is above 4 m/s, the test result compared with the theoretical result is higher. What is more, the greater the belt speed v and vertical load F_v , the more obvious this phenomenon. The reason why this phenomenon occurred may be that when the belt speed v and vertical load F_v are greater, the test apparatus will vibrate severely.

Figure 44.

Test result compared with theoretical result under constant values of R = 44.5 mm and h = 6 mm.

Figure 45.

Test result compared with theoretical result under constant values of R = 54 mm and h = 6 mm.

Figure 46.

Test result compared with theoretical result under constant values of R = 66.5 mm and h = 6 mm.

Conclusion

In this article, a theoretical analysis for rolling resistance based on the viscoelastic property of rubber is given. Then, a total formula of indentation rolling resistance which includes the influence of belt speed v and vertical load F_v on rolling resistance is presented. Finally, using the formula, how the common factors, such as belt speed and vertical load, exert an influence on indentation rolling resistance is analyzed in detail.

An apparatus which can measure the value of resistance is designed. What is more, a comparison between the theoretical results and measured results is performed under different idler radii R. Finally, several conclusions are arrived at through the theoretical analysis and experiments.

From the formula of indentation rolling resistance, belt speed v, thickness of rubber h, idler radius R, viscoelastic property of the backing material of test belt E ₀, E ₁, η ₁, and contact length (a + b) have a direct effect on the value of resistance. However, the load F_v also has an influence on the value of resistance indirectly by exerting influence on the contact length (a + b).

The connection between the belt speed and indentation rolling resistance F_h is closely related. When the belt speed is at the lower level, the decrease in rolling resistance is obvious with the increasing belt speed. However, once the belt speed is above the threshold (6 m/s), the rolling resistance will no longer be changed. It infers that reducing the energy loss is not feasible by way of improving belt speed v. What is more, the threshold of the belt speed v may have a close correlation with viscoelastic property of the backing material of test belt E ₀, E ₁, and η ₁, and thickness of rubber h, radius R, vertical load F_v have little influence on the threshold of the belt speed.

Vertical load F_v exerts an influence on the rolling resistance linearly. Therefore, the load plays an important role in the energy loss of the belt. What is more, it is not to be ignored that the belt speed v plays an important role in the relationship between vertical load F_v and rolling resistance F_h .

Footnotes

Academic Editor: Noel Brunetiere

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) received no financial support for the research, authorship, and/or publication of this article.

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