1/1

Whitepaper

Conveyor Belting

A Comparison of Calculated and Measured Indentation Losses in

Rubber Belt Covers Energy loss due to indentation rolling resistance is a major factor in belt conveyor design. The determination of the indentation rolling resistance of a conveyor belt over the idler system depends on the properties of the rubber compound of the backing, the method of calculation of the resistance value and, of course the conveyor system parameters like carrying weight, idler radius and belt speed. Two theoretical approaches have been employed to predict the loss to be expected for a particular installation and belt cover rubber. The results of these calculations are compared to direct measurements of the indentation rolling resistance in this paper and conclusions offered on important elements of prediction methodologies. Keywords: Indentation rolling resistance, belt conveying (At the time of publication, work on an update to this paper was already in progress. Publication is expected for the beginning of December 2012. Please check back for the update.)

Authors: A.V. Reicks, Overland Conveyor Co., Lakewood (CO), USA

T.J. Rudolphi, Department of Aerospace Engineering, Iowa State University, Ames (IA), USA

C.A. Wheeler, School of Engineering, University of Newcastle, Callaghan (NSW), 2308, Australia

1

A Comparison of Calculated and Measured Indentation Losses in Rubber Belt Covers

A. V. Reicks*, T. J. Rudolphi and C. A. Wheeler Abstract Energy loss due to indentation rolling resistance is a major factor in belt conveyor design. The determination of the indentation rolling resistance of a conveyor belt over the idler system depends on the properties of the rubber compound of the backing, the method of calculation of the resistance value and, of course the conveyor system parameters like carrying weight, idler radius and belt speed. Two theoretical approaches have been employed to predict the loss to be expected for a particular installation and belt cover rubber. The results of these calculations are compared to direct measurements of the indentation rolling resistance in this paper and conclusions offered on important elements of prediction methodologies. Keywords: Indentation rolling resistance, belt conveying

*Allen V. Reicks, Overland Conveyor Co., Lakewood, CO, USA Thomas J. Rudolphi, Department of Aerospace Engineering, Iowa State University,

Ames, IA, USA Craig A. Wheeler, School of Engineering, The University of Newcastle, Callaghan,

NSW, 2308, Australia

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4

 Fig. 1: Schematic of viscoelastic behavior during belt cover indentation 

 

Prediction models are based on various assumptions and consequently may provide different results, even for the same material parameters. One aspect of modeling is that of the kinematics of the backing deformations. Another is how the indentation loss is derived from the deformation model.

Jonkers [3] focuses on the energy dissipation rate of the cover material in a steady deformation cycle whereas Lodewijks [4] and others determine the power of the stress distribution at the idler/backing interface. In the latter case, the effective resistance force of the belt on an idler roll is related to this power through the moment of the interface stress distribution about the center of the idler roll. As most models of indentation rolling resistance are viewed as a steady state process where the cover material moves though a “control volume” region around the idler roll interface, these two different approaches could also be characterized as energy dissipation and moment methods. Alternately, these are internal and external work perspectives. Differences result in these approaches, depending on the material model and the deformation modeling in each, but, of course, under the same assumptions, both the energy dissipation rate and the power methods must provide the same result, assuming no slipping at the idler/backing interface, by the conservation of energy principle.

More complete deformation models such as those of May, et. al., [5] or Hunter [6], treat the backing as fully two-dimensional so that shear deformation occurs in the backing. These more rigorous models generally fall into the category of power methods

x

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5

and ensue from different material and deformation assumptions, and provide some differences in predicted resistance values. Lodewijks [4] has shown that indentation resistance values from these two dimensional approaches are somewhat higher than the one-dimensional models, but not by significant amounts for similar material models.

One can also take a completely computational approach to this rolling contact problem, where the cover deforms as a two-dimensional medium and modeled by finite elements [8]. Like the power method mentioned above, the interface stress distribution and center of reaction offset from the idler roller center is iteratively determined to provide a resistive force on the belt. The advantage of a computational approach is that less restrictive deformation models are possible, such as modeling the entire belt carcass as may be important for cable reinforced belts where the deformation between the steel cables also dissipate energy. On the other hand, recourse to computational methods at the outset does not bring out important parameter dependence or may be time consuming and expensive for parameter studies. 2.1. One Dimensional ‘Winker Foundation’ Models

Modeling of the cover layer as a Winkler Foundation provides a simple yet direct way to analyze the rubber deformation. Though rubber is known to deform in shear, the Winkler model assumes the belt cover to be a bed of independent longitudinal springs so that their deflection versus time can be modeled to conform to an indentation magnitude.

In the approach of Jonkers [3], the strain energy absorbed by the backing material is

approximately determined. Rather than the actual stress/strain path, the load cycle is taken to be that of an elliptical path of the Lissajou oval. The methodology assumes that the deformation cycle experienced by the backing, modeled as a one-dimensional Winker foundation, is continually one of compression followed immediately by tension in a periodic, single frequency, sinusoidal cycle with a half wavelength equal to the contact length with the idler. The load balance – stress equilibrium equation and stress/strain determine the maximum strain o , used to arrive at a ‘correction’ for the rubber

properties. This is described in Rudolphi and Reicks [10] and is used in the results that follow.

Direction of belt motion

Interface stress

Belt

Idler

Fig. 2: Winkler representation of the indentation and stress between the belt and idler.

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h results prov

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tested is glu60 pulse en

a 36 inch torle rotation oeasured acculs can be usemeasure the friction dis

ntation while

6

lso used in tstress, relatinof the idler) mine the actu is establishder the idler

about the idlthe equival

by Rudolphi

ng Resiser indentatioto the wide rr applicationdesign stageg reality to p

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the comparisng the carryand materia

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tance Mon rolling is range of loa

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w.

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7

3.2. The University of Newcastle - Recirculating Belt Test Facility

Tests measuring the rolling resistance of a short belt sample of the same belt cover material were made at the TUNRA Bulk Solids laboratory at The University of Newcastle. The test setup is illustrated in Figure 3. The test facility accepts pre-spliced endless belts up to 600 mm wide and 5500 mm long. The belt speed, idler roll diameter and vertical load can be varied and the influence of each parameter measured independently. The applied vertical load is a result of the vertical component of the belt tension in addition to a minor component due to the self-weight of the belt. The vertical load is varied by adjusting the counterweight. To ensure the contribution of the belt flexural resistance is minimized the deflection of the belt over the instrumented idler roll is limited to conventional sag ratios, with 2.0 % selected for this study.

The total horizontal force , acting on the idler roll is due to the indentation rolling resistance and the rotating resistance of the idler roll. Measurement of the horizontal

Fig. 3: Illustration of the TUNRA Bulk Solids conveyor belt indentation rolling resistance test facility.

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ComparSince the

dictions, resuues at identic

me rubber comThe actual bsame rubbercted. Measufacilities dee made at beperature of ae performed

ds, at the teser compoun

Results olong with caulation metheled M1) anwn in Figureulation meth

quency of gth, which w

aken using aseparate comnstrumentedand externaeach end by

e and rockerd beam, wh

er support faement whicthe idler shapacity load e idler roll.

rison of Ppurpose of ults of the vcal or at leampound andbelt specimer and thicknurements of

escribed in Selt speeds oapproximat

d by the mett temperatu

nd. of the recircualculations mhods assumnd the genere 4 are basehod of Jonk )bav ,

was establish

an instrumenmponent. Td idler roll cally to ensury a collar thar support thaile the horiz

acilitates meh is restricteaft to rotate beam that m

Predictedthis study w

various methst comparabd viscoelasten tested wanesses of 6 mf the indentaSection 3. Mof 1.0 to 5.0 ely 25 oC. Sthods of Secre, idler dia

ulating beltmade by the

me a temperaralized methed on viscoekers, moduluwhere v is hed through

8

nted idler roThis enables consist of a pre concentricat is attacheat enables thzontal force easurement oed only by tfreely abou

measures the

d to Meawas to underhods discussble conditiotic measuremas a solid womm, while aation rollingMeasuremenm/s, at loadSimilarly, cction 2 overameter and m

test facility e two methoature of 25 o

hod of Lodeelastic data tus E was dthe belt spe

h the iterativ

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precision mcity and dynd to the sha

he vertical fois measuredof the verticthe s-type lout the knife-e torque resu

asured Inrstand the ersed above wns, over a raments. oven carcasa 150 mm dig resistance nts on the reds of about 5alculations

r the same ramaterial prop

are shown ods describeoC and incluewijks [4] (lataken at lowdetermined feed and a ve process of

measures thation rolling

made shell whnamic balanaft. Each coforce to be md by a 5.0 kcal force, oad cell. Thedge but is ulting from

ndentatirror and acc

were comparange of norm

ss belt with biameter idleforce were

ecirculating 500 to 2000of the indenange of beltperties from

labeled as Ted in Sectionude that of Jabeled M2)

w strain. Alsfrom the mab is the idlef Lodewijks

he idler rotatresistance t

hich is machnce. The idlollar is suppomeasured bykg s-type loa

while allowhe knife-edgrestricted bythe rotating

on Loss curacy of red to measumal loads an

both coverser roll was taken on thebelt test fac

0 N/m and atntation resist speeds andm the same r

TUNRA in Fn 2. All onkers [3] . All calculso, in the aster curve aer/cover cons’ method.

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Figure

lations

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9

Several observations are immediately clear from Figure 4.

a. Within the range of tested belt speeds, the measured and calculated values of the indentation resistance are nearly independent of the belt speed. This is especially true from the calculated values and fairly well corroborated in the measured values.

b. The calculated values are all considerably less than the measured data and have less load dependency. These represent methods that have been used for conveyor design in the past.

c. The energy dissipation method of Jonkers is somewhat higher than the stress power or moment methods of Lodewijks as expected. (Lodewijks [4]). This is due to the presumption of initial tensile strain inherent to the method.

Based on observation (a) that the indentation resistance is nearly belt speed

independent for this cover compound, further comparisons were performed at the single belt speed of 5.0 m/s. Figure 5 shows the effect of using the strain amplitude corrected material properties as outlined in Section 2, for the Jonkers’ method, M1, and the Rudolphi Reicks adaptation of Lodewijks’ method, M2. Those calculated values, along with the measured values of both tests and the non-strain corrected calculated values, are shown in Figure 5. Note measurements were made at two loads only for the incline roller tests.

0 500 1000 1500 20000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Load [N/m]

Fig. 4: Indentation resistance at low strain and various belt speeds.

Inde

ntat

ion

resi

stan

ce f

acto

r

1 m/s

2 m/s

3 m/s4 m/s

5 m/s

M2

M1

TUNRA test data

10

From the results presented in Figure 5 we further observe that:

a. Low strain and strain corrected prediction results are similar at low load where the strain is low.

b. The strain corrected calculations are higher than those based on non-strain corrected rubber properties and diverge more steeply as the strain effect increases with increasing loads.

c. The 3% strain test data provides parallel but higher loss prediction than those from the low strain data and crosses the strain corrected data at a load representing 3% strain in the backing material.

d. The two measured values with the Inclined Roller test correlate fairly well with the measured values from the TUNRA test facility, but both sets of experimental data are higher than predicted.

e. Even though the strain corrected calculations trend better to the measured values than the non-strain corrected (‘low strain’) results, they are still considerably lower than the measured values.

Point (c) illustrates that use of rubber test data at high strain provides higher loss

predictions which better match test results at intermediate loads but are too high at low loads. Points (d) and (e) lead to the question if indentation rolling resistance may not be the only loss taking place during the testing. In both cases, care was taken to ensure adhesion between the test roll and the rubber was not an important contribution by using talc or a weathered surface as would occur in normal operation. Both had continuous

0 500 1000 1500 2000 2500 3000 3500 4000 45000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Load [N/m]

Fig. 5: Indentation resistance factor - measured and calculated - at various loads.

Inde

ntat

ion

resi

stan

ce f

acto

r

TUNRA test

Roller test

1D M2, low strain1D M2, strain corrected

1D M2, 3% strain, v = 5 m/s

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00

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

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Fig. 6: Inde

Inde

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ion

resi

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acto

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expected loTUNRA testuivalent to anstrumentedly reversed

he magnitudcalculated (

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ow potentialts, additionaa belt deflecd idler roll tostrain cycle

de of this strsee Append

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on is not addese.

Discussiohe use of sevthe convey

rmation in oate results. rrection appltests or by i

1000

stance factor

A test

A test, bendintest

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11

l for added dal rubber deftion to idlero provide the on both sidrain and corrdix A) and p

f Figure 6 prially at low the change ith belt tensielt bending lst versus predressed. Va

n veral methoor belt cove

one dimensio

lied, the comindentation

1500Idler Load [N/

- calculated,

ng corrected

cted

deformationformation dr roll spacinhe radial loades of the beresponding

plotted as a ‘

rovide good load. Note in trend wition and therloss as anothediction sugariation of st

ds for predier runs on idon with the

mparison ofseems to be

2000/m]

measured an

n at the interdue to a consng ratio of 2 ad. This benelt carcass, nviscoelastic

‘corrected’ c

d correlationthe reductio

th load. Theefore roll loher loss mec

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f quantifyinge tolerant of

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n between on in measue latter is duoad. They achanism.

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ReferencJ.D. Ferry: V(1980). T.J. RudolpSME AnnuaC.O. Jonkerapproach; F

hanism. Tha

o found thatnges in the rnearity whented in the cuensional defdergoes the ther assumppredictions

ber deformacover. A so

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er test but iss same phenments.

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phi: Appliedal Meeting ars: The inde

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12

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ng belt bendilies only to ss is due to and also var

n relevant indetailed and er rubber.

of Polymer

elt Cover Lo, Salt Lake ing resistanc4 (1980), p

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f this predicer rolls and ing. It shouthe test casethe real belt

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pp. 312-317.

cted with rea

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ons. f initial cont

that retards ith clean fref talc. It seemost bulk h

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oss predictiaterial mode

on, Wiley, N

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asonable

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es a proper anding of serstood that the measuremtroughed be

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13

4. G. Lodewijks: The rolling resistance of conveyor belts; Bulk Solids Handling, 15, 1 (1995) 15-22.

5. W.D. May, E.L. Morris and D. Atack: Rolling friction of a hard cylinder over a viscoelastic material; Journal of Applied Physics, 30, 11 (1959), pp. 1713-1724.

6. S.C. Hunter: The rolling contact of a rigid cylinder with a viscoelastic half space; Journal of Applied Mechanics, 28 (1961), pp. 611-617.

7. T.J. Rudolphi and A.V. Reicks: Viscoelastic Indentation and Resistance to Motion of Conveyor Belts using a Generalized Maxwell Model of the Backing Material; J. Rubber Chemistry and Technology, June (2006).

8. C.A. Wheeler: Analysis of the Main Resistances of Belt Conveyors; Ph.D. Thesis, The University of Newcastle, Australia, (2003).

9. C.A. Wheeler and P.J. Munzenberger: Predicting the Influence of Conveyor Belt Carcass Properties on Indentation Rolling Resistance, Bulk Solids and Powder: Science and Technology, Vol. 4 (2009) pp.67-74

10. T.J. Rudolphi and A.V. Reicks: The Importance of Nonlinear Strains Considerations in Belt Cover Indentation Loss, Bulk Solids Handling, Vol. 32 (2012).

14

Appendix A Estimate of Rolling Resistance due to Flexure in the Tested Specimen

The Recirculating Belt Test Facility described in Sec. 3.2 controls the lateral belt

load by controlling the belt tension and a small deflection of the belt as depicted in Figure 3. That lateral deflection, though small ( 2%, or 0.04m over a span 2m), introduces a flexural deformation into the test, and thus a contribution to the measured resistance to motion of the belt. The purpose of this appendix is to make an estimate of the flexural contribution to overall measured resistance to motion so that it can be separated from the measured values to give a more realistic comparison of the indentation resistance to calculated values of indentation in Sec. 4.

To quantify the effects of flexure or bending, we take an energy dissipation approach, somewhat similar to that for indentation by Jonkers’ [3] approach for indentation, where it is presumed that the flexural deformation cycle is harmonic, and that as the belt passes over the roller, an amount of energy per unit volume U is dissipated within the belt cover due to hysteresis. The energy dissipated is figured on a half cycle, and is determined by the dynamic rubber properties and the strain amplitude of the longitudinal bending strain o according to the formula [cf. ref. 1],

22 tan22 oo EEU (A1)

Thus, knowing the dynamic moduli, the strain energy dissipated per cycle of

deformation depends on the strain amplitude. Of course this approach has its limitations, stemming mostly from the assumption of the harmonic deformation cycle, which is not the actual case, and is known to overestimate the actual hysteresis loss of a non-periodic cycle, but at least is provides an estimate.

Based on eqn. (A1) and assuming a steady, harmonic deformation process, a general formula for the resistance to motion factor (effective drag force per unit belt width per unit carry load ) that results from equating internal energy dissipation (in a volume of material of unit belt width and half wavelength of the assumed deformation cycle in the direction of belt motion) to the external work (drag force times the half wavelength distance) an equivalent drag force factor is,

z

o dzzzEW

dzzUW

f 2

2

11 (A2)

where z is a coordinate through the belt thickness, and where, in general, the loss modulus E , or equivalently tanE , has been kept within the integration since the dynamic properties may be a function of the strain.

For indentation, Jonkers [3] assumes that the backing behaves as a Winkler foundation (no shear deformation) and presumes the compressive strain through the backing is harmonic in time, or sinusoidal in the distance along the line of contact with the idler with the peak strain amplitude o at the idler centerline. To complete the

15

indentation formula, Jonkers uses the viscoelastic stress/strain relationship and equilibrium of the integrated contact stress with the vertical load on the idler to determine the compressional strain amplitude as,

3134

sin14

cos2

DhE

Wo

where is the effective idler diameter. Then, assuming E or E is not dependent on the strain, and observing that the strain amplitude is independent of (a consequence of the Winkler foundation model), insertion of the above expression for the strain amplitude into the above formula (A2) for if produces Jonkers’ equation for the indentation

resistance factor;

(A3)

We observe here that the drag factor’s dependence on the idler load is ⁄ such that the drag force, as defined as the friction force divided by the normal force ( ), the friction factor for indentation is proportional to ⁄ .

For flexure or bending of the belt, the longitudinal strain through the belt thickness is assumed to be linear through the cross-section according to the “planes remain plane” assumption of simple bending theory such that / , where is the radius of curvature of the belt at the cross-section where the flexural strain is greatest, or where the radius of curvature is a minimum. As in simple beam theory of elastic deformation, the radius of curvature is not a function of z and the drag force for bending, from eqn. (A2) becomes,

z

b dzzEf 220

1

2

(A4)

where again is the radius of curvature at the point of maximum flexural strain in an assumed harmonic deformation cycle. If, as in Jonkers’ formula (A3), we take E to be independent of strain amplitude, the eqn. (A4) becomes,

20

1

2 IE

Wfb (A5)

where I is the second moment of the belt cross-section. To determine bf , it remains to

determine the radius of curvature 0 , which is similar to the determination of the flexural

strain in the harmonic cycle as for the strain amplitude above in the indentation process.

The formula (A5) for the effective drag force due to belt flexure is predicated on a harmonic strain cycle that would be experienced by longitudinal elements of the belt cover as it moves through a half-cycle of continuously recurring cycles of bending. For the situation of interest here, i.e., for the belt testing facility as described in Sec. 3.2, the belt shape through the top test section where the forces on the idler are measured, in steady running conditions, is a complex viscoelastic problem. For purposes here, we approximate the actual shape by solving the static problem of a tensioned beam over half

3431

2 sin14

cos2tan

'2

DE

Whfi

16

the span of the test section, determine the radius of curvature at the mid-span, or at the idler, and presume that the strain at the idler is the maximum, or that the radius of curvature there is a minimum and take the strain to vary harmonically across the top section of the test fixture.

Strain/Radius of Curvature Relationship

To determine the strain amplitude in flexure, or equivalently the radius of curvature , we neglect viscoelastic effects and assume that the belt shape is approximately that of a beam under axial tension T and the governing differential equation for the lateral displacement xw is,

xqEIdx

wd

dx

wd 12

22

4

4

(A6)

where xq is a distributed lateral load, E is the elastic material modulus and I the

second moment of inertia of the cross-section and EIT2 . Corresponding to the test configuration, we solve the homogeneous ( 0q ) differential equation above, with the boundary conditions,

00 dx

dw,

EI

W

dx

wd

20

3

3

, 0Lw , 02

2

Ldx

wd

where W is the vertical (idler) load on the beam at mid-span and L is the distance from the idler to the fixed end pulley.

The solution to the differential equation (A6) is well known, and together with the boundary conditions above determines the deformed shape of the belt due to the lateral load at the idler. From that solution, the lateral displacement at the idler is,

2

1

1

10

23

L

eEI

Wwd

L

(A7)

and the radius of curvature, or inverse of the second derivative of the lateral displacement xw , at the idler,

LEI

W

dx

wd

tanh1

20

12

2

0 (A8)

Then, using the above two equations, given the section modulus , the belt tension (or ) and lateral displacement at the idler, eqn. (A7) determines and eqn. (A8)

determines , or effectively the bending strain, and from eqn. (A4), the resistance factor for bending is,

z

b dzzEEI

LT

f 22 1tanh

1

8

. (A9)

We note here that if E is strain amplitude dependent, and in the presence of the linear strain field for bending, the integration that remains may have to be done numerically. We also observe that if E is taken to be independent of the strain, or not a function of , then the integral would appear as IE , but the I of this calculation might be different than that of the denominator of (A9), as that one is the effective value for the static shape of the belt while the current one pertains to the viscoelastic property of the belt cross-section. These could be different, depending on the belt construction.

17

Sample Calculation

Based on eqs. (A3) and (A9) and the following parameters, the indentation and bending drag forces, as a function of the lateral load were determined.

Test setup parameters: Idler diameter and radius: 150 , 75 . Bottom cover thickness: 6.5 Belt carcass thickness (woven cord): 4.0 Span length from idler to pulley: 1.0 Idler offset at mid-span: 40 Static belt shape parameters: Measured belt section modulus from a cantilever beam/deflection test: 5.78 /

Second moment of the cross-section (solid): 4.094 ∗

10 /

Second moment of the cross-section (no carcass): 2⁄⁄ 4.041 ∗

10 / Effective modulus (measured): ⁄ 5.78 4.094 ∗ 10⁄ 1.412 ∗10 / Viscoelastic belt parameters: Test temperature: 25 Frequency/temperature shift factor (from Fig. 3, Ref. 13): 1.0 ∗ 10 Dynamic moduli (from master curve, Fig. 2, Ref. 13): ′ 2.7 ∗ 10 / , ′′ 2.7 ∗10 / Modulus at test conditions: √ ′ ′′ 2.733 ∗ 10 / ^2 Effective section modulus (used in eqn. A9):

5.78 ⁄ 5.78 2.733 1.412⁄ 11.19 /

The calculations were performed at the various loads as measured in the belt test.

The belt tension that goes into eqn. (A9) was determined by eqn. (7) in an iterative process, i.e., given a load and idler displacement , was determined to satisfy (A7). The results of that calculation for the indentation and bending drag force are shown in Figure A1 below.

18

From Figure A1 we may see that the drag factor for flexure is about 0.0032 across

the range of loads considered, and is actually greater than the indentation drag. However, as the energy dissipation approach tends to overestimate the drag factor, just as seen in the indentation results as calculated by Jonkers’ method (M2) vs. the Lodewijks method (M1) of Figure 4. Thus, instead of taking a drag factor of 0.0032 for the full effect of bending, one can use the ratio of the M2 to M1 curve of Figure 4 to more realistically estimate the contribution of flexure. That ratio of the M2 to M1 curve is about 0.76 over the whole range of loads, so a corrected bending value would be 0.0032*0.76 = 0.0024. To then take into account the flexural effects in the test results, the value of 0.0024 is then subtracted to “correct” the measured TUNRA test results of Figure 5 to arrive at the “TUNRA bending corrected” results of Figure 6.

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

-3

Idler load [N/m]

Fig. A1: Drag resistance factor for indentation and bending by energy dissipation.

Dra

g re

sist

ance

fac

tor

f

Indentation (Eqn. A3)

Bending (Eqn. A9)