Journal of KONE Powertrain and Transport, Vol. 8, No. 0 EFORMLE GROUN INFLUENCE ON THE FRICTION RIVE TRNMIION ETWEEN RIVE PULLEY N ELTOMETRIC TRCK Roert Czaanowski Wrocaw University of Tecnology, epartment of Mecanical Engineering ukasiewicza treet 7/9, 50-37 Wrocaw, Poland tel.:+48 7 30837, fax: +48 7 37645 e-mail: roert.czaanowski@pwr.wroc.pl stract Industrial veicles wit undercarriage wit elastomeric tracks are frequently used in agriculture and forestry, wic is connected wit moving on deformale ground. friction metod of transmitting te drive etween a drive pulley and a track, wic is often applied, is sensitive to adverse amient conditions, wereas te influence of te ground on friction feedack conditions in te area of te ground working on te drive system is significant. eformale ground may e a source of contamination, wic consideraly reduces a friction coefficient on wrap of a drive pulley. Te article includes te analysis of deformale ground influence on te volume of driving moment transmitted etween a drive pulley and elastomeric track. It as een demonstrated tat te influence of deformale ground, as a result of additional pressures working on wrap of a drive pulley may, in some circumstances, consideraly increase te moment transmitted y means of friction feedack of a drive pulley wit elastomeric track. Increase of driving moment may e iger tan 50%. Te analysis includes limited applicaility of Euler and Entelwein s law in te description of transmitting te moment etween a drive pulley wit elastomeric lining and elastomeric track. Keywords: elastomeric track, track undercarriage, operation of tracked veicles, power transmissions. Introduction universal caracter of undercarriage wit elastomeric tracks allows for using industrial veicles equipped wit tem in varied work conditions and on varied grounds. t te stage of designing te drive transmission system, it requires including applicaility area of track weel and steering system. ssuming te use of a macine on deformale ground, te influence of te ground on te undercarriage is presented (Figure ). Fig.. Track undercarriage on te deformale ground [3] Te analysis of power transmission process in a typical track undercarriage of a working macine requires separate consideration of two su-areas of a drive pulley interaction wit te track (Figure ): wrap section free from ground impact ( e ) and wrap section, in wic ground impact must e considered ( g and ).
R. Czaanowski Te analysis of drive transmission process was conducted on te track undercarriage, wose first roller at te same time is a tension pulley, and te last roller serves as a track sprocket. Te selection of suc a structure is justified wit more advantageous force distriution in te track maximum track tension is oservale only witin a sort section of te track [3].. rive transmission process analysis etween drive pulley and elastomeric track nalysing te researc results presented in te literature [4, 5], it as een noted tat witin te drive transmission system under consideration, pressure is te most significant parameter influencing a friction coefficient value and, terefore, te following dependance etween a friction coefficient and pressure as een derived []: p were: a, approximation coefficients of experimental researc results, p pressure. a ln, ().. Te coupling of te drive pulley wit te track on wrap eyond te influence of te soil Te forces acting on a separate element of track are presented in Figure. From equilirium equations and considering tat sin(d/)= d/ and dsin(d/)0 we otain: dp p d p 0 () and considering () we otain: dp p d a lnp 0. (3) pr d M p +d d 0 Fig.. Te forces acting on a separate element of track [] Equation (3) a nonlinear first order differential equation, wose result is as follows: p e e C a. (4) Considering oundary condition (p(=0)=p ), on sustitution to equation (4) we otain: ln aln p p exp e a. (5) Te aove equation enales calculation of unit pressures value in eac point of a elting curve. pplying dependance: p=/( ), it is possile to define force in te track in any point of wrap: 0
a e p z p V a ln ln exp. (6) driving moment transmitted y friction feedack witout ground impact (eg. veicle moving on ard ground) may e calculated (for veicle moving forwards) witin equation: exp ln ln 0 V a a e M V, (7) were: M drive moment, diameter of drive pulley, widt of drive pulley, V force in te tensioning device, 0 angle of wrap, a, approximation coefficients, (eq. ). Te formula (7) is not continuous for force in a tension device equalling zero ( V =0). It is caused y an accepted form of aproximation function (). Calculations applying equation (7) must e conducted for unit pressures in te area of approximate experimental results. Fig. 3. Comparison of te results of measurements () wit calculations ased on: ()-equation (7), (3)-Euler s law (μ=08 )[] Figure 3 presents a drive moment transmitted to friction feedack depending on initial tension of elastic and. In calculations, friction coefficient values were used as well as function coefficents caracteristic for identical working conditions. s it can e seen (figure 3), a course of moment calculated from Euler formula is consideraly different from te one oatined experimentally. Totally different caracter of ot graps is particularly noticeale. It is different wit calculations conducted according to te elaorated model. However, tere are differences as to te value altoug te caracter of ot graps is similar, wic appears to confirm te selection of te metod of considered feedack coefficient variation.... rive pulley feedack wit te track in te impact area of deformale ground Ground deformation may e resilient or plastic in its nature. In te latter case, deformation may e caused y means of eiter density or floating of te ground. Ground deformation is mainly dependent on ground type (granulation) and ground condition (porosity, umidity). Te simplest dependence etween pressure eigt working on te soil and soil squeeze is [7]: n z k p, (8) eformale Ground Influence on te Friction rive Transmission etween rive Pulley and Elastometric Track
were: p unit pressure on te ground, z dept of soil sinkage under load, k soil deformaility module, n dimensionless index defining soil structure and type. Te studies [] demonstrated tat parameter n does not depend on load area size, ut on soil type and its omogeneity. Parameter k is dependent not only on ground type and condition, ut also on geometrical types of stancion. Terefore, ekker [] divided coefficient k into two modules of soil vertical deformation: dependent on its coesion k c and dependent on its internal friction k : k c p k z n, (9) were: widt of stancion. Many autors (eg. Reece [6]) attempted to develop formulas (8) and (9). ccepting a similar model and accepting Terzagi s equation [8] as a asis: po c NC N z Ng, (0) were: c coesion, - specific gravity of soil, N c, N, N g coefficients, Reece [6] suggests: n z p c k' c k'. () ependences presented aove: pressure-sinkage, are adequate only in te case of original ground load. Wen considering a track veicle moving on deformale ground, it is necessary to e aware of ground eaviours in case of unloading and reloading, wic is oservale wen susequent rollers of te track undercarriage move. Te course of an exemplary curve loadingunloading-reloading is presented in Figure 4. Wong [9] suggests, in first approximation (for sections and C, figure 4), te following linear dependence: p p k z z, () were: p pressure at te eginning of unloading, z sinkage at te eginning of unloading, k parameter during i-t unloading-loading cycle. Tis assumption is possile, since participation of ysteresis in total deformation is negligile, and wit sandy ground, it disappears completely. Is as een determined tat te value of coefficient k during unloading and reloading canges along wit canging sinkage z : k R. Czaanowski were: k o and E are soil parameters otained troug experiments. o E k z, (3)... nalytical assessment of deformed ground impact on te volume of driving moment transmitted Movement of veicle wit track undercarriage on deformale ground, on account of drive moment transmitted, as two significant aspects (Figure 5): additional pressures in te tangential area of pulley on te ground fragments g and, increase of wrap area g. imension of wrap under consideration g and witin te ground impact area depend on te range of significant parameters. Te most significant ones are as follows: type and condition of deformale ground, drive pulley widt and diameter, moving veicle velocity,
eformale Ground Influence on te Friction rive Transmission etween rive Pulley and Elastometric Track ground deformation velocity, loading istory (multi-pass-effect), type and properties of undercarriage elements, track initial tension, pysical and mecanical properties of elastomeric track and tread form, drive pulley vertical loading. Fig. 4. Response of mineral soil for cyclic loading [9] Fig. 5. Interaction of a drive pulley and elastomeric track wit pressures on deformale ground ( e wrap range witout ground impact, g - additional wrap resulting from ground deformation troug a drive pulley, wrap range wit relaxation of te soil)[] etermining values g and on an analytical asis is igly complex and requires elaorating track undercarriage model including interaction wit te ground. On account of tis issue complexity, impact interaction assessment of a drive pulley wit deformale ground on te moment transmitted volume is limited to: increase of wrap angle wit additional angle g, influence of unit pressures from te ground in wrap angle area, werey volume (at te same time z ) is a function of elasticity properties of te soil including loading istory (multipass-effect). ependence etween elastic reflection z and is as follows z =0,5 (-cos( )), ence, angle =arcos(-z / ). nalogously z =0,5 (-cos( g )) and g =arcos(-z / ). Figure 6 presents te increase of driving moment transmitted, caused y wrap angle increase wit additional angle g. Calculations were conducted using equation (8), te influence of additional unit pressures from te ground on te track were not included (data for calculations: =0,8 m, =0,76 m, V =80 kn, 0 = + g rad, a=.809, =-0.734 [/ln(pa)]). In order to include te influence of pressures from te ground, te section of te track -x was suject to toroug analysis (Figure 6). Pressures distriution working on te separated track element in tis area is presented in (Figure 7). From equilirium equations in te radial and tangential directions, considering tat: sin(d/)d/, cos(d/), dsin(d/)0 and applying te following dependences: dr=μdn, =(p) and p=p(z) upon transformations, te following first order differential equation was otained: pplying te following dependence: d d r pz p 0. (4) z cos, (5) r 3
R. Czaanowski 7.5 5 M were z' ' 0 M [knm].5 0 7.5 5 M were z' ' 0 Movement direction e d dn.5 z'' g g 0 0.0 0.04 0.06 0.08 z'' [m] x +d dr r d p r d Fig. 6. Increase of driving moment M caused y wrap angle increase wit additional angle g =f(z ) [] Fig. 7. istriution of pressures acting on te track element from te tangential area of a drive pulley wit deformed ground [] were: angle coordinate eginning from, functions p=p(z) troug p=p() and (p)= () were verified: d d r p. (6) On account of relatively low values of angles g and and minor value canges of feedack coefficient μ=f(p), te aove equation (6) is solved for μ=const. Te result of omogeneous equation: d 0 (7) d is: a C e. peculiar result was otained wen introducing anticipated equation result (4): C e (9) on differentiation: was otained: (8) ' C' e C e and sustituting to equation (4) te following e C e C e r p C'. (0) On transformations, te following was otained: e r p e d. () ince ()= a ()+ () te following final result was otained: e C r p e d. () Recent considerations include neiter tickness nor structure of te track (reinforcement lines). Neverteless, wit more precise analysis of tangential tensions influence from te ground on te generated feedack force, it may e noticed tat tangential force on te external track surface and tread r d (Figure 7) causes elastomeric layers deformation, ut only to lines surface separating te track section, wose lengtening, on account of low values of tangential force and ig rigidity of lines, is negligile. Terefore, te influence of tangential forces from te ground 4
eformale Ground Influence on te Friction rive Transmission etween rive Pulley and Elastometric Track does not reac te tangential spere of te track and drive pulley and does not influence feedack process significantly. Tus, equation () was solved witout tangential tensions impact ( 0 ). Te aove equation was solved as first witin te range g-x. Introducing pressure-sinkage function into equation (): p z k z', (3) te following was otained: e C r k r cos z e d. (4) Linear equation (8) (n=) was accepted on account of inconsiderale ground deformations (practically witin resilience range). ividing te integral of te aove equation into te sum of integrals: r cos e d z' e d. (5) Introducing te solved integrals ( and ) into equation () te following was otained: r e C r k e cos sin z' C C 3, (6) taking: C=C +μ r (C +C 3 ) te following was otained: r C e r k z' cos sin. (7) Constant C was determined ased on te oundary condition (=0)= and te following was otained: r e rk e cos sin cos sin z' e. (8) In order to calculate a moment transmitted troug feedack, it is necessary to calculate force in te elt in point (taking = ): r e r k e cos sin z' e, (9) force in point ( ) was solved taking 0 =-. Total driving moment (including te deformed ground impact may e calculated from te elow formula: ' M r rk exp e V e ln aln a e e cos sin z' e V. (30) Taking data from previous example (Fig. 6) for calculations and applying equation (30), te drive moment transmitted was calculated including vertical ground reactions (Figure 8). 3. Conclusions Te aove analyses indicate tat vertical ground reactions may exert significant influence on te driving moment transmitted troug drive system increasing it even y 50%. Tis increase may e iger if eyond spere, te second area of cooperation of a drive pulley wit 5
R. Czaanowski deformale ground g will is considered. Comparale pressure values in curve spere g cause at least te same (or iger) increment of te moment transmitted in comparison to curve spere g. Fig. 8. riving moment as function of drive pulley sinkage and force in a tension device [] Terefore, total increment of te transferred drive moment caused y ground impact may e even twice as muc iger tan increment otained in te aove calculations, tus ground vertical reactions may cause te transmitted drive moment increase even y 00%. References [] ekker, M., G., Off-te-Road locomotion. Researc and development in terramecanics, nn ror Te University of Micigan Press 960. [] Czaanowski, R., Modelling of coupling of running gear wit elastomer track, issertation, Wrocaw University of Tecnology, Wrocaw 00. [3] udziski, P., New Ideas for moile track system wit elastomer elt. Part, Off-Road Macine and Veicles in Teory and Practice, Proceedings of te st International Conference Ed. y Piotr udziski []. ITV East European Office, Tecnical University of Wrocaw, eptemer 3 4, Wrocaw 996. [4] Homann, R., Zur Kraftüertragung von ntriestrommeln auf Fördergurte mit talseileinlagen, issertation, TH Hannover 975. [5] Pfleger, P., Pampel, W., Ergenisse der Reiwertermittlung zwiscen ntriestrommel und talseilfördergurt, Heezeuge und Fördermittel, 3/980. [6] Reece,. R., Principles of soil-veicle mecanics, Proc. f te Institution of Mecanical Engineers, d. 80, Pt Numer, 965/966. [7] otyski,., Mecanics of soil-veicle system, MON, Warszawa 966. [8] Terzagi, K., Teoretical oil Mecanics, Jon Wiley and ons, New York 959. [9] Wong, J., Y., Terramecanics and Off-Road Veicles, Elsevier 989. 6