外文翻譯--滑輪牽引驅(qū)動皮帶輸送機 英文版

satisfactory match and that belt speed has little eect on traction in the feasible speed range.the carrying strand of the belt without interfering with the bulk material flow on the belt, it cannot take fullthat occur in each section. This is the key to keeping the tension in the belt low, giving the opportunity to*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 Theory0094-114X/$ - see front matter C211 2006 Elsevier Ltd. All rights reserved.advantage of the benefits a distributed drive system has to oer.An alternative drive method, which oers 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 resistancesC211 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 alongTraction 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 adoi:10.1016/j.mechmachtheory.2006.01.005A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 13361345 1337use the same light belt construction regardless of the overall conveyor length. This results in reduced belt costs,greater layout flexibility and oers 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 coecient 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 dierence with a drive pulley the Eulerequation 2, which is normally used to determine the maximum transferable eective 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 ofwrap angledrive pulleytroughedconveyor beltsupport rollcontact lineFlvbFcclamping rollerdrive wheelFlpouchconveyor belttriangular profilesupport rollers contact patchvbbulk materialFig. 1. Support and drive configuration of a conventional (left) and a pouch (right) belt conveyor.the 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 matchbetween 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, sucesfor 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 stiness Eiand a dashpot with a damping coecient gi, as illustrated in Fig. 3.Thewhereelementwhere1338 A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 13361345strains 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 patchxFzri giC1 _ei； 3beEand egrepresent the local strain of the spring and dashpot elements, respectively. The sum of the localri EiC1 eEi； 3agr0 E0C1 e. 2The stress in the remaining spring and dashpot elements is directly related to the local strains of the individuali1r0is the stress in the single spring, which is directly related to the material strain eMaxwell element, orr r0Xmri； 1total stress in this model is equal to the sum of the stress on the single spring and the stresses in eachIdeally 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.Fig. 2. Rigid cylinder rolling on a curved visoelastic surface._eEi_riEi； 5a_egirigi. 5bto sinusoiin the1E1E02E2mEmFig. 3. Modelling the viscoelastic properties with Maxwell elements.A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 13361345 1339ulus E00and loss factor tand. Together they represent the complex modulus of elasticity and are related asfollows:dal varying stresses and strains 8,9. Fig. 4 shows the results of such experiments for the rubber usedEBS belt. The results of these experiments are typically expressed as the storage modulus E0, loss mod-Combining 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 dierential 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 subjectedFig. 4. Measured and approximated viscoelastic properties.functionstressesperiodicmodulusapproximTothe deformassumptribution.x C28 Rcontactness hbe writt1340 A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 13361345and the deformation equation (12) e h, the dierential equation (6) for each Maxwell element canen asoriC0 riEiC0Eix. 13wx； yz0C0x22R1C0y22R2with z0c22R2. 12Under steady state conditions with a constant belt speed vbdxdtC0vb, using a Winkler foundation with thick-wx；ysolve this equation, the pressure distribution r(x,y) in the contact plane is determined first, by definingation of the viscoelastic surface in the direction of the z-axis (see Fig. 2). For this calculation antion, also used by Johnson 3, is made that the shear stress does not influence the normal stress dis-If the contact zone is small compared to the curvatures of the rolling cylinder and rubber surface (so1and y C28 R2), and the cylinder is pressed into the surface with a distance z0, then the deformation of thesurface can be described as follows:generally 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 oftheoptimisation routineduringthematchingprocedureincreasinglydicult.Furthermore,dueto the implemented least squares approach, the maximum number of elements is physically limited by theamount of experimentally measured data. It isimpossible 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 dierent num-bers of Maxwell elements are used. The figure clearly illustrates the dierence 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 coecient.of elasticity in a desired frequency range. With a possible operational belt speed of 1.610 m/s and anated contact length of 0.02 m, the frequency of excitation ranges from 80 to 500 Hz. The accuracyE0 E0Xmi1x2g2iEix2g2i E2i； 9E00Xmi1xgiE2ix2g2i E2i. 10The number of Maxwell elements m to be used in the model depends on the required accuracy of the complexof the model parameters and the excitation frequency x. This is accomplished by eliminating theof the individual Maxwell elements from Eq. (1) with Eq. (6) and substituting the strain e with thefunction sin(xt), which results intan d E00E0. 8To fit the properties of the model onto the measured data the storage and loss modulus are expressed as aEC3 E0 i C1 E00； 7ox givbhR1This dierential 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； yE02R1ha2C0 x2Xmi1EikihR1x C0 a a ki1C0 expx C0 akiC18C19C18C19C18C19with kigivbEi. 14The resulting normal force Fzcan now be calculated by integrating the stress distribution over the whole con-tact region orFzZcC0cZayC0byrx； ydxdy. 15The trailing edge of the contact plane positioned at C0b(y) is found by setting r(x,y) equal to zero.4. Shear stress distributionapparenb 1where x is the angular velocity of the drive wheel.ThedA.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 13361345 1341vbhingezxh Fzbrush elementR1oxC0h. 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 eects. 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.Mcreep ratio is related to the shear angle by the following equation:oc dd vbjj； 16t velocity is also know as the creep ratio d and is defined as follows:v C0 xRWith the calculated pressure distribution and a measured friction coecient, 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 dierenceor creep does occur between the drive wheels outer diameter and the belt when a traction force is applied. ThisFig. 5. Brush model.ByG, shearelements.To derubberIf it iswhere5. Correctis smaltact region,wherewherefrom1342 A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 13361345has to match the creep curve described by Eq. (23), which is calculated bylimd!0F0t fsZaC0bsstickdy fsdhg02a b2 vbXni1gia b C0 kig1C0 exp C0a bkigC18C19C18C19C18C19 !； 25g vwhereE is the statically measured modulus of elasticity. With this equation the normal force F0zis eliminatedEq. (23). To match the stiness of the brush model, the tangent at the start of the models traction curveThe 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 8F0zR1C0 m2pEr； 242RF0ztaF0tand F0zare measured per unit length of the contact width.responding limit for the creep ratio, as derived by Johnson 3 using a half space approximation, isd aF0tor F02RF0z； 23stiness of the layer. Under the condition that the speed dierence between the drive wheel and the beltl, the slip region at the trailing edge becomes vanishingly small. As there is virtually no slip in the con-the occurring speed dierence or creep is predominantly determined by the layer stiness. The cor-A correction factor fsis introduced to compensate for the fact that the Winkler foundation does not incor-porate the shearing eect between adjacent spring elements and to match the stiness of the model with theactualion factorthe shear stress reaches the friction boundary and it can be found by solvingsstickt1； yl C1 rt1； y. 22t1(y) represents the transition line separating the stick from the slip-zone. It represents the edge whereequation (18) can be found, yielding the shear stress in the stick-zonesstickx； ydhG0a C0 xXmi1dgivbh1C0 expGix C0 agivbC18C19C18C19C20C21. 20The contribution of both the stick and slip-zone can now be calculated by integrating the calculated shearstress in each zone separatelyFtractionZcC0cZt1yC0byl C1 rx； ydx Zayt1ysstickx； ydx !dy； 21G 21 m. 19assumed that the stick-zone starts at the leading edge of the contact plane, a solution to dierentialexperiments were available, the shear parameters were derived from the normal stress experiments and con-verted with the aid of the following equation:Erive the viscoelastic shear parameters, additional oscillatory experiments should be conducted where thetest sample is subjected to shear stresses and strains. However, due to the fact that no results of sheardescribing the shearing of each Maxwell element can be written asosioxC0 siGigivbC0Gidh. 18replacing the modulus of elasticity E, stress r and strain e in Eq. (1), (2) and (6) with the shear modulusstress s and shear angle c respectively equations are derived that describe the behaviour of the brushUnder steady state conditions and using the deformation equation (17) the dierential equationkigi bgiand fsis the correction factor.Elimination of F0tby combining Eqs. (23)(25) gives the following correction factor:fsapEh41C0 m2p；p g02a b2 vbXni1gia b C0 kig1 C0exp C0a bkigC18C19C18C19C18C19:26The stiness 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,AtcompenbrakeA.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 13361345 1343adjustable springzFdwhere 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 dierent 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 alrR1C01C0FtlFzs !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, dierent 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 wheelFMdhingeMbthe start of each experiment the contact force and the drive wheel speed are set to a desired value. Tosate for a decrease in brake wheel diameter due the indentation of the rubber layer, the speed of thewheel is adjusted just below synchronous speed until the brake torque reduces to zero. From this point,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.Fig. 6. Experimental layout with drive and brake wheel.1344 A.J.G. Nuttall, G. Lodewijks / Mechanism and Machine Theory 41 (2006) 133613450 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16050010001500slipTraction (N) Maxwell model Elastic half space x MeasurementsFz = 500 N Fz = 1000 N Fz = 1500 NFig. 7. Comparison of experiments and model (vb= 1.6 m/s).The 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 eect 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 r