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Traction versus slip in a wheel-driven belt conveyorA.J.G. Nuttall*, G. LodewijksDelft University of Technology, Transport Technology and Logistics, Mekelweg 2, 2623 CD Delft, The NetherlandsReceived 13 July 2005; received in revised form 15 December 2005; accepted 2 January 2006Available online 2 March 2006AbstractThis paper presents an extension of existing models, used for flat belt conveyors, to describe the relationship betweentraction and slip in a wheel-driven belt conveyor with a curved surface. The model includes the viscoelastic properties ofthe rubber running surface in the form of Maxwell elements. After the application of a correction factor to account for theinteraction between adjacent elements, which is initially not modelled, experimental results show that the model generates asatisfactory match and that belt speed has little effect on traction in the feasible speed range.? 2006 Elsevier Ltd. All rights reserved.Keywords: Rolling contact; Traction; Viscoelasticity; Maxwell model; Pouch belt conveyor; Curved belt surface1. IntroductionTraditionally belt conveyors for transporting bulk material have a drive station at the head and/or tail ofthe system where the belt is wrapped around a drive pulley, see Fig. 1. It is a well proven drive configurationfor belt conveyor systems with a single or dual drive stations. However, problems arise when more than twodrive stations are desired. Due to the fact that the drive pulley cannot be placed at any arbitrary location alongthe carrying strand of the belt without interfering with the bulk material flow on the belt, it cannot take fulladvantage of the benefits a distributed drive system has to offer.An alternative drive method, which offers greater layout flexibility in a multiple drives system, is to imple-ment drive wheels that directly press onto the belts surface to generate the desired traction force. In theEnerkaBecker System (abbreviated EBS) for example motors with drive wheels mounted on their outputshafts form a drive pair that can be placed at virtually any location along the belt. Bekel 1 also proposedvulcanising a drive strip to the bottom of a conventional flat conveyor belt, so it could be driven by a pairof drive wheels. The freedom to place the drive stations at any location along the belt gives the system designeran opportunity to control the tension in the belt by balancing the installed drive power with the resistancesthat occur in each section. This is the key to keeping the tension in the belt low, giving the opportunity to0094-114X/$ - see front matter ? 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2006.01.005*Corresponding author. Tel.: +31 15 2782004; fax: +31 15 2781397.E-mail address: A.J.G.Nuttallwbmt.tudelft.nl (A.J.G. Nuttall).Mechanism and Machine Theory 41 (2006) 13361345/locate/mechmtMechanismandMachine Theoryuse the same light belt construction regardless of the overall conveyor length. This results in reduced belt costs,greater layout flexibility and offers possibilities to standardise the system components.For both the conventional drive pulley and drive wheel, like in the EBS, the generated traction force isdetermined by the friction coefficient and the contact force between the belt and the pulley or drive wheel sur-face. However, with the drive wheel configuration the contact force is not primarily determined by the belttension, but by the normal force that is generated as a result of the weight of the belt and the bulk solid mate-rial it carries and the force generated by a clamping roller. Due to this difference with a drive pulley the Eulerequation 2, which is normally used to determine the maximum transferable effective traction in a conven-tional belt conveyor, cannot be applied to a wheel-driven conveyor. Therefore, a new model needs to be for-mulated that takes the material and geometric properties of the belt and drive into account.This paper presents a model that describes the relationship between traction and slip in the rolling contactpatch of a wheel-driven belt conveyor like the EBS. The model includes the viscoelastic rubber properties ofthe rubber belting material as an array of Maxwell elements and is compared to an elastic approach used byBekel 1. Both models are also compared with experimental results. The tractionslip relationship is of interestbecause the traction and slip combined with the applied normal contact force greatly influence the wear rate ofthe belts surface. To prevent the belt from wearing out before its guaranteed lifetime, a maximum allowablewear rate has to be set, which can result in a derating of the maximum transferable traction.2. Modelling contact forces based on viscoelastic propertiesA number of researchers have used the Maxwell model to quantify the energy dissipation of a cylinderrolling on a viscoelastic surface 35, which is comparable to a conveyor belt passing over an idler. As thecover passes the idler the rubber surface compresses and relaxes in quick succession. Due to the viscoelasticproperties of the rubber cover material the relaxation will take some time. This causes an asymmetrical stressdistribution that results in a resistance force. To incorporate the viscoelastic properties and calculate theindentation resistance, the Maxwell model has mainly been used in its three parameter form. One model inparticular, described by Lodewijks 6, combines the three parameter Maxwell model with a Winkler founda-tion or mattress consisting of springs that do not interact with each other. Because shear forces betweenadjacent spring elements are not considered calculations become less complex. Despite the simplificationresults show that this representation of the belt cover behaviour generates satisfactory results. Therefore,the combination of the Maxwell model and Winkler foundation will serve as starting point for the analysisof the relationship between traction and slip of a wheel-driven conveyor belt.In order to adopt a similar approach to describe the traction force exerted by a drive wheel in the EBS, themodel is expanded in two ways. Firstly, the number of Maxwell elements is increased to accommodate a matchwrap angledrive pulleytroughedconveyor beltsupport rollcontact lineFlvbFcclamping rollerdrive wheelFlpouchconveyor belttriangular profilesupport rollerscontact patchvbbulk materialFig. 1. Support and drive configuration of a conventional (left) and a pouch (right) belt conveyor.A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 133613451337between the model and the real rubber behaviour throughout the contact patch. Secondly, a brush model, alsoused to describe the rubber tread behaviour of car tyres 7, is introduced to calculate the shear forces causedby slip between the drive wheel and the belt.The three parameter Maxwell model, consisting of a single Maxwell element in series with a spring, sufficesfor a conventional conveyor belt because the contact surface between the belt and idler can be described by aline contact. With a constant contact length throughout the contact zone the model only has to match for asingle frequency of excitation, making a good approximation possible by tuning the time constant of the singleMaxwell element to this frequency. However, as a result of the curved running surface in the EBS, there existsan elliptical contact zone. Due to the varying contact length in the elliptical patch, the model has to match fora range of frequencies. Fig. 2 shows how the model represents the belt passing over an idler or drive wheel. Arigid cylinder rolling with angular velocity x is pushed onto a curved viscoelastic surface moving with the beltvelocity vb, which results in the elliptical contact patch.To match the model with the rubbers viscoelastic properties within the excitation range, additionalMaxwell elements are introduced. An array of Maxwell elements approximates the viscoelastic behaviour eachconsisting of a spring with stiffness Eiand a dashpot with a damping coefficient gi, as illustrated in Fig. 3.Ideally this model would have an infinitely large number of elements. However, due to practical and compu-tational reasons the ideal situation is simplified by limiting the number of elements to m.The total stress in this model is equal to the sum of the stress on the single spring and the stresses in eachMaxwell element, orr r0Xmi1ri;1where r0is the stress in the single spring, which is directly related to the material strain er0 E0? e.2The stress in the remaining spring and dashpot elements is directly related to the local strains of the individualelementri Ei? eEi;3ari gi? _ egi;3bwhere eEand egrepresent the local strain of the spring and dashpot elements, respectively. The sum of the localstrains is equal to the total strain on the element. Taking the time derivative of the strains leads to_ e _ eEi _ egi.4The time derivatives of eEand egcan be found from the Eqs. (3a) and (3b)zyvbR2R1curved running surfacerolling cylindercontact patchxFzFig. 2. Rigid cylinder rolling on a curved visoelastic surface.1338A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 13361345_ eEi_ riEi;5a_ egirigi.5bCombining Eqs. (5a) and (5b) with Eq. (4) results the following relationship between the total strain and thestress in each spring dashpot assembly_ ri riEigi Ei_ e.6Together with Eq. (1) and (2) the differential equations (6) of all Maxwell elements form a set of equations thatwhen solved gives the normal stress in the contact plane.The parameters of the Maxwell model have to be tuned to match its complex modulus of elasticity withviscoelastic properties of the belt cover measured in oscillatory experiments where the material is subjectedto sinusoidal varying stresses and strains 8,9. Fig. 4 shows the results of such experiments for the rubber usedin the EBS belt. The results of these experiments are typically expressed as the storage modulus E0, loss mod-ulus E00and loss factor tand. Together they represent the complex modulus of elasticity and are related asfollows:1E1E02E2mEmFig. 3. Modelling the viscoelastic properties with Maxwell elements.Fig. 4. Measured and approximated viscoelastic properties.A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 133613451339E? E0 i ? E00;7tan d E00E0.8To fit the properties of the model onto the measured data the storage and loss modulus are expressed as afunction of the model parameters and the excitation frequency x. This is accomplished by eliminating thestresses of the individual Maxwell elements from Eq. (1) with Eq. (6) and substituting the strain e with theperiodic function sin(xt), which results inE0 E0Xmi1x2g2iEix2g2i E2i;9E00Xmi1xgiE2ix2g2i E2i.10The number of Maxwell elements m to be used in the model depends on the required accuracy of the complexmodulus of elasticity in a desired frequency range. With a possible operational belt speed of 1.610 m/s and anapproximated contact length of 0.02 m, the frequency of excitation ranges from 80 to 500 Hz. The accuracygenerally increases when more elements are added. However, with more elements the model also becomes morecomplex, making computations more time consuming and the search for starting conditions that give a goodconvergence of the optimisation routine during the matching procedure increasingly difficult. Furthermore, dueto the implemented least squares approach, the maximum number of elements is physically limited by theamount of experimentally measured data. It is impossible to fit a model with more parameters than data points.Fig. 4 shows how the model fits onto the measured viscoelastic properties of the EBS when different num-bers of Maxwell elements are used. The figure clearly illustrates the difference between the simplest model withone element (or three parameters) that gives an unsatisfactory approximation between 10 and 1000 rad/s and amodel with three elements (or seven parameters) with an improved accuracy. The seven parameter model wasfinally chosen as a good match and used for further calculations.3. Normal stress distributionWhen a drive wheel applies a traction force to the conveyor belt within the traction limit, stick and slip-zones exist in the contact plane. In the stick-zone only the rubber surface deforms due to the applied traction,while in the slip-zone the rubber surface also slides over the wheels surface because the friction limit has beenreached. To determine the placement of the zones, friction is modelled according to the Coulombs dAmon-tons law:jsx;yj 6 lrx;y;11where l is the friction coefficient.To solve this equation, the pressure distribution r(x,y) in the contact plane is determined first, by definingthe deformation of the viscoelastic surface in the direction of the z-axis (see Fig. 2). For this calculation anassumption, also used by Johnson 3, is made that the shear stress does not influence the normal stress dis-tribution. If the contact zone is small compared to the curvatures of the rolling cylinder and rubber surface (sox ? R1and y ? R2), and the cylinder is pressed into the surface with a distance z0, then the deformation of thecontact surface can be described as follows:wx;y z0?x22R1?y22R2with z0c22R2.12Under steady state conditions with a constant belt speed vbdxdt ?vb, using a Winkler foundation with thick-ness h and the deformation equation (12) e wx;yh, the differential equation (6) for each Maxwell element canbe written asoriox? riEigivb ?EixhR1.131340A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 13361345This differential equation can be solved by setting the pressure at leading edge a(y) of the contact plane equalto zero or r (a,y) = 0 because at the first point of contact no deformation has occurred yet. Solving the equa-tion reveals the pressure distribution in the contact planerx;y E02R1ha2? x2 Xmi1EikihR1x ? a a ki 1 ? expx ? aki?with kigivbEi.14The resulting normal force Fzcan now be calculated by integrating the stress distribution over the whole con-tact region orFzZc?cZay?byrx;ydxdy.15The trailing edge of the contact plane positioned at ?b(y) is found by setting r(x,y) equal to zero.4. Shear stress distributionWith the calculated pressure distribution and a measured friction coefficient, most of the information isavailable to determine the shear stress within the slip-zone, as determined by Eq. (11). The next essential stepto find the shear stress distribution in the whole contact plane is the calculation of the shear stress in the stick-zone.In the stick-zone no sliding takes place between the contact surfaces. However, an apparent speed differenceor creep does occur between the drive wheels outer diameter and the belt when a traction force is applied. Thisapparent velocity is also know as the creep ratio d and is defined as follows:d vb? xR1vbjj;16where x is the angular velocity of the drive wheel.The creep ratio is related to the shear angle by the following equation:ocox ?dh.17To establish a relationship between the creep ratio and shear stress distribution in the stick-zone, the Maxwellmodel is combined with a brush model that describes shearing effects. The brush model depicted in Fig. 5 is asimplified representation of the belt cover in the contact region. It consists of rigid elements that hinge and areheld in place by a torsion spring at their base. The behaviour of the torsion spring is also based on the Maxwellmodel analogous to the spring element in Fig. 3.zxR1vbFzhMdhingebrush elementFig. 5. Brush model.A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 133613451341By replacing the modulus of elasticity E, stress r and strain e in Eq. (1), (2) and (6) with the shear modulusG, shear stress s and shear angle c respectively equations are derived that describe the behaviour of the brushelements. Under steady state conditions and using the deformation equation (17) the differential equationdescribing the shearing of each Maxwell element can be written asosiox? siGigivb ?Gidh.18To derive the viscoelastic shear parameters, additional oscillatory experiments should be conducted where therubber test sample is subjected to shear stresses and strains. However, due to the fact that no results of shearexperiments were available, the shear parameters were derived from the normal stress experiments and con-verted with the aid of the following equation:G E21 m.19If it is assumed that the stick-zone starts at the leading edge of the contact plane, a solution to differentialequation (18) can be found, yielding the shear stress in the stick-zonesstickx;y dhG0a ? x Xmi1dgivbh1 ? expGix ? agivb?.20The contribution of both the stick and slip-zone can now be calculated by integrating the calculated shearstress in each zone separatelyFtractionZc?cZt1y?byl ? rx;ydx Zayt1ysstickx;ydx !dy;21where t1(y) represents the transition line separating the stick from the slip-zone. It represents the edge wherethe shear stress reaches the friction boundary and it can be found by solvingsstickt1;y l ? rt1;y.225. Correction factorA correction factor fsis introduced to compensate for the fact that the Winkler foundation does not incor-porate the shearing effect between adjacent spring elements and to match the stiffness of the model with theactual stiffness of the layer. Under the condition that the speed difference between the drive wheel and the beltis small, the slip region at the trailing edge becomes vanishingly small. As there is virtually no slip in the con-tact region, the occurring speed difference or creep is predominantly determined by the layer stiffness. The cor-responding limit for the creep ratio, as derived by Johnson 3 using a half space approximation, isd aF0t2RF0zorF0t2RF0za;23where F0tand F0zare measured per unit length of the contact width.The normal force F0tcan be expressed as a function of the distance to the leading edge a of the contact zone.Bekel 1 derived the following equation, using the Hertz formulas:a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8F0zR1 ? m2pEr;24where E is the statically measured modulus of elasticity. With this equation the normal force F0zis eliminatedfrom Eq. (23). To match the stiffness of the brush model, the tangent at the start of the models traction curvehas to match the creep curve described by Eq. (23), which is calculated bylimd!0F0t fsZa?bsstickdy fsdhg02a b2 vbXni1gia b ? kig1 ? exp?a bkig? !;25where kiggivbgiand fsis the correction factor.1342A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 13361345Elimination of F0tby combining Eqs. (23)(25) gives the following correction factor:fsapEh41 ? m2p;p g02a b2 vbXni1gia b ? kig1 ? exp?a bkig?:26The stiffness of the model is compensated by scaling the Maxwell parameter with the factor of Eq. (26).6. Experimental validationExperiments were conducted to measure the actual relationship between traction and slip at a drive stationin the EBS and validate the presented model. During the experiments two wheels were used, see Fig. 6. Onewheel made from steel represents the drive wheel and is driven by an electric drive motor. The other wheel,representing the belt cover, has a rubber layer (h = 30 mm) vulcanised to it. It is also connected to an electricmotor that is used as an adjustable brake. Strain gauges on each motor shaft measure the produced torque. Anadjustable spring was also used to pull the brake wheel onto the drive wheel, making it possible to control thecontact force. The diameters (Dd= Db= 500 mm) of both wheels were chosen such that their contact patch,created when pressed against each other, is comparable with the patch between the drive wheel (D = 250 mm)and the belt in the EBS.At the start of each experiment the contact force and the drive wheel speed are set to a desired value. Tocompensate for a decrease in brake wheel diameter due the indentation of the rubber layer, the speed of thebrake wheel is adjusted just below synchronous speed until the brake torque reduces to zero. From this point,where the measured traction is zero, a traction slip curve is created by successively decreasing the brake wheelspeed and measuring the resulting increasing traction. Fig. 7 presents the results for different contact forcesand a constant speed. It also shows the curves that were calculated with the presented viscoelastic modeland the equations presented by Bekel that he used to describe the traction slip relationship for a wheel-drivenrubber strip 1. Bekel used a similar half space approach as described by Johnson 3 for a line contact involv-ing completely elastic material, which results ind alrR1 ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ?FtlFzs !with1rR1r11r2.27The results show that the presented Maxwell model gives a good match with the measured values for low con-tact forces. As the contact force increases the model starts to underestimate the actual traction.To assess the influence of the viscoelastic properties on traction, different curves where calculated with vary-ing speeds. Fig. 8 presents the results for a constant contact force with speeds ranging from the EBSs stan-dard belt speed of 1.6 m/s to a potential high speed application with a belt speed of 10 m/s.rubber layerbrake wheeldrive wheelFzMdFdhingeMbadjustable springFig. 6. Experimental layout with drive and brake wheel.A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 133613451343The curves in Fig. 8 suggest that traction decreases with increasing speed, with the greatest reduction occur-ring in the middle part of the slip range. However, this effect seems very small in the feasible speed range of abelt conveyor. With the speed influence in the same order of magnitude as the measurement error, it can beconcluded that in this case the viscoelastic part of the rubber properties has a small influence on the relation-ship between traction and slip.7. ConclusionThis paper shows that it is possible to expand a three parameter Maxwell model, which is used to calculatethe rolling resistance of a cylinder rolling on a viscoelastic layer, and include the behaviour required to deter-mine the relationship between traction and slip. Such a model has a number of simplifications, making itrelatively simple and computationally friendly for a contact model that includes viscoelastic behaviour. Afterthe introduction of a
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