Viscoelastic Indentation and Resistance to Motion of Conveyor Belts using a Generalized Maxwell Model of the Backing Material

Transcription

1 Vscoelastc Indentaton and Resstance to Moton of Conveyor Belts usng a Generalzed Maxwell Model of the Backng Materal homas J. Rudolph a and Allen V. Recks b a Department of Aerospace Engneerng, Iowa State Unversty, Ames, IA b Desgn Manager, Overland Conveyor, Inc., Denver, CO Abstract A one-dmensonal Wnkler foundaton and a generalzed vscoelastc Maxwell sold model of the belt backng materal are used determne the resstance to moton of a conveyor belt over dlers. he vscoelastc materal model s a generalzaton of the three-parameter Maxwell model that has prevously been used to predct the effectve frctonal coeffcent of the rollng moton. Frequency, or loadng rate, and temperature dependence of the materal propertes are ncorporated wth the tme/temperature correspondence prncple of lnear vscoelastc materals. As a consequence of the Wnkler foundaton model, a normalzed ndentaton resstance s ndependent of the prmary belt system parameters carryng weght per unt wdth, dler dameter and backng thckness - as s the case for a three-parameter vscoelastc model. Example results are provded for a typcal rubber compound backng materal and belt system parameters. Keywords: Rollng contact; Vscoelastc layer; Generalzed Maxwell model; Wnkler foundaton; Indentaton resstance Rudolph/Recks //6

2 Introducton Advances n rubber materals, belt fabrcaton and conveyor system technology, together wth demands for larger systems to transport bulk granular materals, put a premum on effcent system desgn. he power to drve large conveyor systems s a crtcal desgn parameter, due to energy costs over the lfe of the system. Dsspatve energy losses due to nelastc deformaton and ndentaton of the belt coverng, or backng, materal on a belt as t passes over each dler of the system s generally consdered to be one of, f not the, domnant loss mechansm n the system. Hager and Hntz estmate ths source of power loss to be as much or greater than 6% of the total. Predctve analytcal models of ths loss are thus mportant to desgners of such systems and several models have been developed and used. he model of rollng resstance developed heren follows and bulds on the work of several authors, ncludng May, Morrs and Atak, Jonkers, Spaans 4, Lodewjks 5,6 and Johnson 7. hese authors model the ndentaton of the dler nto the backng as a onedmensonal stress/stran stuaton wth the dler consdered rgd and the backng as a Wnkler foundaton on a rgd base. hus the compressed fbers, through the depth of the backng, act ndependently such that there s no accountng for shear n the deformaton process. he full two-dmensonal models of Hunter 8 and Morland 9 account for shear deformaton, although they presume the backng to be a vscoelastc half-space, as opposed to havng a fnte depth, and requre the soluton of dual ntegral Equatons to satsfy the contact equlbrum and deformaton condtons. More mportantly, however, n the analyses mentoned above, the vscoelastc materal behavor s presumed to be that of a smple, three-parameter Maxwell sold. However, rubber materals commonly used for backng on conveyor belts are hghly temperature and load-rate dependent, so a useful vscoelastc consttutve model must accurately account for these effects. he three parameter model s extremely range lmted, n both temperature and rate effects, to capture real vscoelastc behavor. In the present analyss, the three-parameter consttutve model s generalzed to arbtrary + parameters, and used n combnaton wth the Wnkler foundaton deformaton model. hermal and rate effects are ncorporated through the well-accepted tme/temperature superposton prncple 9, of lnear vscoelastcty to provde a smple, yet suffcently comprehensve model of the deformaton and materal response, to capture realstc materal behavor over a broad range of temperatures and load rates. he Indentaton Resstance Model For a lnear vscoelastc materal and a one-dmensonal state of stress, the stress relaxaton formula for a prescrbed stran hstory s determned by, + () t = Ψ() t ε ( ) + Ψ( t s) ε ( s) t σ & ds () where Ψ () t s the stress relaxaton functon, & ε dε dt s the stran rate and σ () t s the stress at tme t. For the present analyss the tme t = s taken to be that nstant at ε =. whch the belt frst contacts the dler so that ( ) Rudolph/Recks //6

3 v Backng materal z δ x ( x, z) h x = b x = a ω R Idler Fgure. Indentaton of the dler nto the backng materal and geometrc parameters. Followng the work and general notatonal conventons of Jonkers, Spaans 4, Lodewjks 5,6 and Johnson 7, the knematcs of the deformaton process for the belt backng passng over an dler at unform speed v s shown n Fgure. At a constant speed the deformaton process s essentally steady state wth respect to the Euleran coordnate x = a vt, where t denotes tme and a the pont of contact of the belt and dler ( x s fxed wth respect to the dler, but not rotatng wth t). he problem of steady moton may then be cast n terms of the coordnate x as opposed to the explct tme t through the above transformaton. For a small maxmum ndentaton depth δ compared to the dler radus R, or δ R <<, the ndentaton amount at a pont x on the surface of the dler s approxmated by a parabola such that z = δ x R, where δ = a R. Hence the compressonal stran of the backng materal at the coordnate x, modeled as a Wnkler foundaton or ndependent lnear elements through the backng, s z ε = = ( δ x R) = ( a x ) () h h Rh Usng the translatng coordnate speed relatonshp, x& = v, so z & = xx& / R = xv / R, the stran rate, or tme dervatve of Equaton (), becomes &ε = z & h = vx Rh. Wth ths stran rate, the compressonal stress between dler and belt, from Equaton (), may be expressed as a functon of t or as a steady functon of x, Rudolph/Recks //6

4 t v ξ x σ = Ψ( t s) ( a vs) ds = Ψ ξ dξ () Rh Rh a v ow, for a gven materal relaxaton functon Ψ, ths Equaton can, n prncple, be ntegrated to explctly determne the contact stress profle between the belt and the dler as depcted n Fgure. However, the contact length a + b, or both the pont of frst contact a and the pont of departure b, are undetermned; they depend on the load or net x σ ( x) z x = b x x = a R Idler Fgure. Contact stress dstrbuton between dler and backng. force between the belt and dler, and hence, as n most stuatons nvolvng contactng materals, ths problem s geometrcally nonlnear. At equlbrum, the load W (force per unt wdth across the dler) and the resultant of the stress dstrbuton must be n balance such that a b ( x) W = σ dx (4) o determne a and b, Equatons () and (4) must be satsfed smultaneously, but an explct soluton s not possble. However, as shown by Lodewjks 5, a smple teratve procedure can be used to balance the load and the contact length, determnng both a and b, gven the load W. Once the stress profle and contact length s determned, the moment M of the stress dstrbuton about the center of the dler s Rudolph/Recks //6

5 a b ( x) M = xσ dx (5) hen the resstance force to rollng s M R and the coeffcent of rollng resstance, or equvalent Coulomb drag coeffcent µ, s µ = M RW = R a b a xσ b ( x) dx σ ( x)dx In the followng secton, the above Equatons are evaluated, ncorporatng the generalzed materal model, to provde a closed form model for the ndentaton resstance. he Generalzed Maxwell Materal Model For rubber compounds used as backng for belt conveyors, t s common to use the smple three-parameter Maxwell consttutve model. For realstc backng materals, however, the three parameter model s not capable of representng vscoelastc behavor over a range of temperatures and loadng rates. he three parameter model s easly generalzed to + parameters as shown n Fgure, or to what s sometmes called the Zenar (6) η η η L η L η E E E E E E L L Fgure. he mechancal elements of the Maxwell sold model. model. he constants E and η represent the varous elastc and dsspatve vscous elements of the model and the constant E s referred to as the long term or equlbrum modulus. Accordng to ths generalzed + parameter model, the stress response functon s gven by the Prony seres t / τ Ψ ( t) = E + Ee (7) = where τ = η / E are the characterstc perods of the varous dsspatve elements of the mechancal model. Inserton of ths response functon nto Equaton () for the stress, t = a x /, one has, ncludng the transformaton ( ) v Rudolph/Recks //6

6 x ( ξ x) / vτ E + Ee Rhσ ( t) = ξ dξ (8) a = hen, smlar to the -parameter model, each term of seres n ths Equaton can be ntegrated to arrve at the result, ( ) = + + x E Rh = x x + k a x σ ( x) Ek k e (9) a a a a where k = vτ / a are the Deborah (wave) numbers correspondng to the element pars of the materal model. ow, ntroducng the non-dmensonal length parameter ς = b / a, and equatng the stress of Equaton (8) to zero at x = b provdes the condton, E ( + ς ) k ( + ς )( ς ) + Ek [ k ς ( + k ) e ] = () = For gven materal parameters, the soluton of ths Equaton would provde only the rato ς = b / a, not a or b explctly. he addtonal condton that must be satsfed s Equaton (4), whch, when ntegrated and also expressed n terms of ς s Rh E ( + ς )/ k W = ( + ς ς ) + Ek ( ς ) k [( + k )( e ) ( + ς )] () a 6 = hen, for gven materal parameters E, k, and belt system parameters W, R and h, the values of ς and a, whch satsfy both Equatons () and (), wll be n equlbrum wth the load W. he smultaneously soluton s readly accomplshed numercally by a smple teratve procedure: (a) Intalze a guess for ς and solve Equaton () for a. (b) Use ths value of a to determne the materal parameters k = vτ / a. (c) Wth these values of k and E, solve (numercally) Equaton () for ς. (d) Use the updated value of ς, replace the ntal guess of step (a) and contnue repetton of steps (a), (b) and (c) untl ς converges to a predetermned accuracy. Followng ths teratve process to determne an accurate value of ς = b a, a and b, Equaton (5) can then be evaluated to get the moment of the stress dstrbuton. hus, 4 Ea 4 M = ( ς + ς ) + 8R h () 4 Ea k k ( + ς )/ k k ( + ς ) + ( + ς ) k ( + k )( k + ς ) e = R h he ndentaton resstance coeffcent then follows from Equatons (6), () and (). Combnng these Equatons n such a way as to elmnate the explct appearance of the contact length parameter a, Equaton (6) can be wrtten as / M Wh M R W E ( ) 4 / R µ = = () F where the non-dmensonal functons F and M have been ntroduced and are gven by, Rudolph/Recks //6

7 F M E ς (+ ) / k ( ς ) ( + ς ς ) + k k [( + k )( e ) ( + ς )] 6 4 ( ς ) ( ς + ς ) + 8 = E E k k = E k ( + ς ) ( + ς ) + ( + ς ) k ( + k )( k + ς ) e In both these functons, the modul have been normalzed to ( ) E ς (4) / k E, gvng rse to the rato E n each expresson and ntroduces the non-dmensonal factor ( E ) / R coeffcent of µ n Equaton (). Equaton () then shows the drect dependence of µ on ( Wh R ) / (5) Wh as a, as s a drect consequence of the Wnkler foundaton model and consstent wth the results of the three parameter materal models of other -6. Incluson of E n ths factor renders t nondmensonal. Reflectng ths factor, a normalzed ndentaton factor F s defned by, Wh M F µ = E ( ) 4 / R (5) F It s noted that snce ths normalzed ndentaton factor s the rato of two terms that depend only on the materal parameters k and E, so after the load dependent factorς has been determned by the teratve process descrbed above, then F s solely a functon of the backng materal vscoelastc propertes. As such, F s ndependent of the belt system parameters W, h and R, unlke the coeffcent µ, s essentally a materal property from whch the ndentaton resstance can be determned for any W, h and R. / emperature Dependent Materal Modelng Mechancal propertes of a vscoelastc materal are typcally measured by cyclcallymposed deformaton tests on standard specmens n smple extenson, shear, bendng or torson at controlled temperatures and at relatvely low frequences to avod nertal effects. By testng over a wde temperature range but lmted frequences, and usng the superposton prncple, whch relates tme (equvalently frequency or speed) and temperature effects, one can extrapolate to frequences or load rates consderably beyond those of the test. hs theoretcal rheologcal model for vscoelastc materals, supported by expermental measurements, and postulatng the equvalence of temperature and tme effects, s known as the tme/temperature superposton prncple 9,. Accordng to ths prncple, f denotes temperature and a reference temperature, then a materal property at tme t and temperature s equvalent to that at tme a t and temperature, where a s a shft parameter n tme so that a = a (, ). A commonly used expresson for a, applcable to amorphous polymers at temperatures above the glass transton temperature g, due to Wllams, Landel and Ferry and referred to as the WLF Equaton, s the logarthmc form, log( a ) = C( ) [ C + ( )], where C and C are constants determned emprcally from data. Rudolph/Recks //6

8 Alternately, for data taken from cyclcal or frequency controlled stran experments, an emprcal representaton of the tme/temperature shft parameter a may be determned by overlayng, through shfts n the frequency, the temperature dependent modulus data onto a master curve. he requred frequency shft, on a logarthmc scale, for data at each test temperature, to produce the overlay determnes the equvalent of the WLF Equaton, or may be taken the basc temperature/frequency relatonshp of the superposton prncple. hs master curve approach s used here to relate temperature and frequency, or load rate effects, wthn the generalzed Maxwell materal model. So nstead of the above functonal form for a of the WLF Equaton wth only two constants, the relatonshp between a and temperature s taken as, n n ( a ) a = a + a + a + L+ a log = n (6) = and the constants a are determned by a least squares curve fttng process n the constructon of the master curve for the materal. Expermental measurement of the mechancal propertes of polymerc materals s most often done by subjectng specmens to cyclc stran programs at fxed temperatures and over a lmted range of frequences. For a snusodal stran hstory of the form ε = ε sn( ωt), where ω s the crcular frequency and ε the ampltude, and after an ntal transtory state, the stress follows the stran n frequency, but at a delayed phase. For elastc materal, the stress and stran are n phase, but for vscoelastc materals there s a delayed response, and t s convenent to express ths relatonshp n the form, / σ ( t) = E ( ω) ε ( t) = [ E ( ω) + E ( ω) ] ε sn( ωt + δ ) (7) where E s the magntude of the complex modulus wth real and magnary components E ( ω) and E ( ω), called the storage and loss modul, respectvely, and δ ( ω) = E ( ω) / E ( ω) s the phase angle or lag of stress behnd stran. For the generalzed Maxwell model, the storage and loss modul are related to the mechancal elements of the model shown n Fgure by, ω τ E ( ω ) = E + E (8) = + ω τ and ωτ E ( ω ) = E (9) = + ω τ where E are now the dscrete or spectral values of the Maxwell model and τ = η E are the perods of the varous elements. Implct n materal propertes E ( ω) and E ( ω) s the dependence on temperature. In the frequency doman, the superposton prncple relates propertes at temperatures and frequences accordng to the relatonshps E ω, = E a ω,, E ω, = E a ω () ( ) ( ) ( ) (, ) From these Equatons, whch represent the materal s master curves of ( ω) ( ω) E and E, one can extrapolated to frequences, and hence loadng rates, consderably Rudolph/Recks //6

9 beyond those of the test data. For a gven belt speed v, the relatonshp between load rate and frequency s taken to be v ω = π () a + b whch derves from the presumpton that an element of the materal of the backng passes through a half-cycle of loadng durng the contact perod. hs Equaton, whch couples the materal propertes and contact length, and hence the materal propertes to the deformaton and equlbrum Equatons, through the teratve process outlned prevously for Equatons () and () n the establshment of a and b. Fgures 4 and 5 show the combnaton master curves and shft factor of the shear modul G ( ω) = E ( ω) and G ( ω) = E ( ω) for a typcal, ncompressble, belt backng materal. he tests were performed wth a Rheometrc Scentfc ARES mechancal spectrometer n a twstng/shear mode on flat specmens wth approxmate dmensons of o 5xx4mm. Data were taken test over a temperature range of 6 to + 7 o C at about o ncrements and frequences from. to 5 Hz wth about equally spaced ncrements on a logarthmc scale. he shft factor a of Fgure 5 was developed accordng to Equaton (6), from the data at each temperature ncrement, startng from a chosen reference temperature. Storage (G') and Loss (G'') Modul [Pa] Storage (G') and Loss (G'') Modul vs. Frequency Shfted G' data Shfted G'' data Master poly (n = ), G' Master poly (n = ), G'' G' from spectrum G'' from spectrum Frequency [Hz] Fgure 4. Master curve for a typcal belt backng materal. Rudolph/Recks //6

10 he reference temperature for ths curve s o C, whch s where a equals of Fgure 5, and a polynomal of order 5 s shown on the curve wth the dscrete a values of the fttng process, as prescrbed by Equaton (6). 5 emperature/frequency Shft Factor vs. emerature Least squares data ft Polynomal (n = 5) ft 5 a emperature [C] Fgure 5: Frequency/temperature shft factor. For ths materal, as characterzed by the master curve and frequency/temperature shft factor, the dscrete + parameters of the Maxwell model,.e., values of E and η of Equatons (7) and (8) are also determned. hs amounts, equvalently, to the determnaton of the dscrete spectrum of the materal, where Equatons (8) and (9) represent the dscrete versons of comparable contnuous transforms. Determnaton of these dscrete spectral values has been the focus of much research and varous technques and mathematcal fttng processes schoegl and Emr and schoegl 4. eural networks and genetc algorthms have also been employed to best ft both E and η, or equvalently E and τ, from gven data as n Fgure 4. Alternately, a common and farly smple approach s to fx the perods τ = ω at values equally spaced on the log ( ω) scale (half-decade ncrements s suffcent ) and then use a least square curve ft to the data as calculated from the polynomal ft of the master curve at the τ = ω values over the range of the data. Here, a non-lnear, non-negatve, least square method s used to ft the data. he non-negatve constrant on the fttng Rudolph/Recks //6

11 process s not necessary, but elmnates the possblty of negatve E coeffcents, whch s not uncommon for ths type of transform. Fgure 6 shows the resultng spectral values that result from fttng the G master curve to Equaton (8). he re-calculate values of G and G that result from the spectrum, through Equatons (6) and (7) are shown n the orgnal master curves n Fgure 4. ote that the re-calculated G s accurate, snce the spectrum s ft to ths data, but also the calculaton of G from the spectrum derved from fttng G s also 9 Dscrete Spectrum of Storage Modulus G' 8 7 log(spectral Strength) [Pa] log(τ [s]) Fgure 6. Dscrete spectrum of the storage modulus G. accurate over the range of the data. Some devaton s evdent at the frequency extremes, but the consstency of the ft s evdence of the redundancy nvolved n the spectral characterzaton of both Equatons (6) and (7). Indentaton Resstance Calculatons Wth the dscrete spectrum of Fgure 6 as a characterzaton of a typcal backng materal, the ndentaton resstance model of Equatons () (5) were evaluated for a range of temperatures and belt speeds, and at typcal values of system parameters: W = lb / n = 75. / m, h =.5 n =.65 m R = n =. 76m. Belt speed, beng a functon of tme, relates also to temperature through the superposton prncple, Rudolph/Recks //6

12 whch allows for the constructon of unversal curves of µ and F, for any temperature/speed combnaton when graphed as functons of va. hus, Fgures 7 and 8 gve, respectvely, the frcton coeffcent µ of Equaton () and the non-dmensonal value F of Equaton (). 4.5 x - Frcton Coeffcent vs. emperature Scaled Belt Speed 4.5 µ - frcton coeffcent va (m/s) Fgure 7: Indentaton resstance µ vs. scaled belt speed. Rudolph/Recks //6

13 . ormalzed Frcton Coeffcent vs. emperature Scaled Belt Speed F = µ/(wh/e R ) / va (m/s) Fgure 8. ormalzed ndentaton resstance F vs. scaled belt speed. hese graphs are actually compostes, or overlappng segments, of the calculaton of µ and F through the algorthm above, wth the calculatons performed through nested loops through ncrements n the temperature and speed v. When graphed aganst va, these resultng curves actually overlap, but the overlappng portons are only fantly vsble n the graphs, whch s a consequence of the superposton prncple. hus, Fgures 7 and 8 are unversal curves, mplct n temperature,.e., applcable to all temperatures n the range provded by the shft parameter a of Fgure 5. Determnaton of specfc values of µ or F from the curves of Fgures 7 and 8, at gven values of the belt speed v and temperature, s then done n conjuncton wth the a curve of Fgure 5. Gven, one would frst fnd the value of a from Fgure 5, provdng the ordnate va for the correspondng value of µ and F from Fgures 7 and o 8. hus, for ths materal at a temperature of, say, = C, Fgure 5 provdes a value a, and at a speed of m/s so that va = m/s, the values of µ and F from Fgures 7 and 8 would be located on the curves just at about the peaks of the curves and be about. 44 and. 89, respectvely. As noted prevously, utlty of the nondmensonal F of Fgure 8 s that t pertans to systems wth any values of W, h and R. For the same materal and calculatons, Fgure 9 shows the correspondng contact length parameter ς = b / a as results from the teraton process at whch the load W s n equlbrum wth the contact stress. It s noted here that a mnmal value of ς occurs at Rudolph/Recks //6

14 va m/s, whch, not surprsngly, s the same speed/temperature at whch the ndentaton resstance factors µ and F attan ther maxmum values. b/a vs. emperature Scaled Belt Speed.95.9 ζ = b/a va (m/s) Fgure 9. Contact length rato vs. scaled frequency. Dscusson and Conclusons A predctve model of the ndentaton and rollng resstance of a lnear vscoelastc materal backng on a conveyor belt usng a generalzed Maxwell materal model of the backng materal has been made. hs represents a generalzaton of the three parameter model that has tradtonally been used for the calculaton of the ndentaton resstance, but whch s capable of capturng the real polymerc behavor of rubber backngs used on large, modern conveyor systems. he vscoelastc layer s presumed to be one dmensonal n the sense of that of a Wnkler foundaton. Explct formulae for the resstance coeffcent were developed and sample calculatons wth materal propertes typcal of a common backng materal were made. he partcular materal consdered n the example calculatons shows the typcal peakng of the ndentaton resstance at some pont n the temperature/speed range. As an ad to desgn and mnmzaton of the power requred for conveyor systems, the developed methodology s drectly usable for the evaluaton of the performance of the backng materal, once the materal propertes are establshed through standard testng methods. Rudolph/Recks //6

15 References M. Hager and A. Hntz, he energy-savng desgn of belts for long conveyor systems, Bulk Solds Handlng,, 4 (99), pp W.D. May, E.L. Morrs and D. Atack, Rollng frcton of a hard cylnder over a vscoelastc materal, Journal of Appled Physcs,, (959), pp C.O. Jonkers, he ndentaton rollng resstance of belt conveyors: A theoretcal approach, Fördern und Heben,, 4 (98), pp C. Spaans, he calculaton of the man resstance of belt conveyors, Bulk Solds Handlng,, 4 (99), pp G. Lodewjks, he rollng resstance of conveyor belts, Bulk Solds Handlng, 5, (995) G. Lodewjks, Desgn of hgh speed belt conveyors, Bulk Solds Handlng, 9, 4 (999), pp K.L. Johnson, Contact Mechancs, Cambrdge Unversty Press, Cambrdge, UK, S.C. Hunter, he rollng contact of a rgd cylnder wth a vscoelastc half space, Journal of Appled Mechancs, 8 (96), pp L.W. Morland, A plane problem of rollng contact n lnear vscoelastcty theory, Journal of Appled Mechancs, 9 (96), pp A.S. Wenman and K.R. Rajagopal, Mechancal Response of Polymers: An Introducton, Cambrdge Unversty Press, Cambrdge, UK,. J.D. Ferry, Vscoelastc Propertes of Polymers, rd Edton, Wley, ew York, 98. M. Wllams, R. Landel and J. Ferry, he temperature dependence of relaxaton mechansms n amorphous polymers and other glass-formng lquds, Journal of the Amercan Chemcal Socety, 77 (955), p. 7.. W. schoegl, he Phenomenologcal heory of Lnear Vscoelastc Behavor: An Introducton, Sprnger, Berln, I. Emr and.w. schoegl, Generatng lne spectra from expermental responses. Part I: Relaxaton modulus and creep complance, Rheologca Acta, (99), pp. -. Rudolph/Recks //6