The Mechanics of Tractor Implement Performance

Transcription

1 The Mechnics of Trctor Implement Performnce Theory nd Worked Emples R.H. Mcmilln CHAPTER 6 HITCHING AND MECHANICS OF THE TRACTOR CHASSIS Printed from: CONTENTS 6.1 INTRODUCTION IMPLEMENT HITCHING Introduction Hitching systems 6.3 () Triled - one point hitch 6.3 (b) Semi-mounted - two point hitch 6.3 (c) Fully mounted - three point hitch TRACTOR CHASSIS MECHANICS Centre of grvity 6.5 () Longitudinl loction 6.5 (b) Verticl loction Issues 6.8 () Weight trnsfer 6.8 (b) Instbility Anlysis nd ssumptions WEIGHT TRANSFER Four wheel trctor Weight trnsfer with rolling resistnce Weight trnsfer with hitching systems 6.21 () Anlysis 6.21 (b) Comprison of hitching systems Other emples 6.28 () Two wheel (wlking trctor) 6.28 (b) PTO driven triler 6.31 (c) Triled implement weight trnsfer system IMPENDING INSTABILITY REFERENCES 6.40 Note: The Title Pge, Prefce, Tble of Contents, Inde, Appendices nd detils of the Frmlnd trctor cn be found with Chpter 1.

2 6.1 INTRODUCTION 6.1 CHAPTER 6 HITCHING AND MECHANICS OF THE TRACTOR CHASSIS It ws shown in Chpters 4 nd 5 how the weight on the wheels of trctor determines its trctive force nd rolling resistnce, hence its drwbr pull nd trctive efficiency. This weight depends on: (i) the sttic forces, viz, * the weight of the trctor * tht prt of the implement weight (if ny) tht is crried by the trctor (ii) the effect on the trctor of the dynmic forces rising from the ction of the implement, viz, * drught (horizontl) force(s) * verticl force(s) In designing nd using the trctor - implement system, it is desirble to tke dvntge of ll these forces to increse (nd control) the weight on the trctor wheels while still ensuring the stisfctory performnce of the trctor nd the implement. For given optimum weight on the wheels, the more tht is provided by the dynmic effects, the less tht hs to be provided by the sttic weight. The three-point linkge system introduced by Ferguson, which mde significnt use of the dynmic forces on the implement to provide weight on the driving wheels, llowed the introduction of very light trctor. This feture is now used on most smll to medium sized trctors. Before considering the mechnics of the trctor chssis we need to review the methods of hitching (ttching) implements to the trctor s these hve significnt influence on how the implement forces determine the dynmic weight on the trctor wheels. The following gives brief review of those spects of implement hitching tht re relevnt to the performnce of the trctor. Other detils of the vrious systems my be found in the references t the end of this Chpter. 6.2 IMPLEMENT HITCHING Introduction The hitching of implements nd the mechnics of the chssis my be studied by considering two perpendiculr plnes: (i) (ii) the verticl longitudinl plne down the centre line of the trctor in which we consider the symmetricl forces such s the weight, the wheel rections nd the direct effect of the implement forces. the horizontl plne where the moment effect of the implement forces which re not symmetricl (eg, unsymmetricl or off-set implements nd ll drft forces in turning) will ffect the ttitude nd steering of the trctor. These influence the opertion of the trctor but re not relevnt to the norml (stright hed) performnce of the trctor; they will not be considered further in this book.. The hitching of implements to trctors my be mde in vrious wys nd plces. For this purpose the trctor hs one or more stndrd ttchment loctions t the rer nd for some trctors t the front, in the form of: (i) (ii) linkges for 'djustble' ttchment; djustment in the verticl plne is usully mde by mens of n inbuilt hydrulic (hydro-sttic) pump driven by the trctor engine. drwbrs for 'fied' ttchment; djustment is mde mnully or with 'eternl' or 'remote' hydrulic cylinders supplied with oil from the in-built hydrulic pump in the trctor. The stndrd hitching systems my be clssified s follows. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

3 6.2 () (b) Figure 6.1: Triled (one point) implement hitches () without nd (b) with verticl force. Figure 6.2: Semi-mounted hitch where the front of implement is crried on horizontl pivot. Figure 6.3: Fully mounted, rer three-point linkge hitch. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

4 Hitching systems () Triled - one point hitch Here the implement is ttched to the trctor t one (drwbr) hitch point. This represents the simplest rrngement, but it provides minimum in the wy of implement control nd weight trnsfer. The implement, which is usully crried on wheels (for support nd / or depth control), is free to move in both the horizontl nd verticl plnes s it follows the vrying ground surfce. Two common rrngements cn be identified. (i) (ii) where the implement is fully crried on its wheels nd its drwbr is pivoted t both ends; the implement force is essentilly horizontl, Figure 6.1 (). where the front of the implement (such s in n unblnced triler or similr two-wheeled implement) is crried on the trctor drwbr nd the rer on wheel or wheels, Figure 6.1(b). There is usully significnt sttic verticl component in the implement ttchment force nd hence the weight trnsfer from implement to trctor rer wheels is greter thn in (i) bove. The triled hitch is lest effective in terms of both weight trnsfer nd implement control when compred with other systems (see Section 6.4.3). The former wekness hs been overcome by the development of weight trnsfer hitch for triled implements in which prt of the weight of the implement nd / or the downwrd soil forces re supported by the trctor rer wheels. This system is considered in Section 6.4.4(c). (b) Semi-mounted - two point hitch In this rrngement the front of the implement is crried on the lower links of the trctor nd the rer on cstor wheel s in Figure 6.2. In the verticl, longitudinl plne the implement is free to pivot bout the outer ends of the lower links nd hence it behves s the one point hitch bove, ie, it is free to follow ground undultions. It is, however, rigid in the horizontl plne nd is therefore frequently used for un-symmetricl implements hving side forces, such s mouldbord or disc ploughs, or offset drught forces, such s forge mowers. There is usully significnt sttic verticl component in the implement ttchment force becuse prt of the weight of the implement nd of the downwrd soil forces re supported by the trctor. Thus weight trnsfer would be greter thn in corresponding triled implement; see Section (c) Fully mounted - three point hitch Here the implement is ttched to the trctor by mens of the three-point linkge s shown in Figure 6.3. In this side view the lower two points re coincident; the upper point is midwy between, but bove the lower two. This system totlly constrins nd llows complete control of the implement. It is not free to swing in spce like the triled implement, nor in the verticl plne like the semi-mounted; it must operte in the position determined for it by the linkge. The eception to this sttement is tht the implement is usully free to rise, ie, it is not held down by the linkge. If it does rise, it will be due to the upwrd soil forces being greter thn implement weight; it will, however, move in wy determined by the kinemtics of the linkge. In the verticl longitudinl plne (Figure 6.3) the linkge hs the form of mechnism known s 'four link chin', the chrcteristics of which re treted in books on kinemtics. We cn identify the four links s shown in Figure 6.4: (i) the two lower links (which ct s one in the verticl plne) (ii) the upper or top link (iii) the implement frme or pedestl (iv) the trctor chssis. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

5 v 3 1 Figure 6.4: Three-point linkge s four br 'chin'. Line of soil force () V Virtul hitch point nd instntneous centre of rottion V (b ) V (c) Figure 6.5: Three point linkge s n implement is lowered () ner commencement of penetrtion (b) in 'stble' free link condition (c) restrined bove the free link condition The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

6 6.5 The significnt point is 'v' t the intersection of the upper nd lower links. When discussing the motion of the implement it is termed the instntneous centre of rottion ; t the instnt shown, the implement moves s if it ws rotting bout tht point. The point 'v' itself moves from instnt to instnt, hence the motion of the implement is quite comple. When discussing the forces on the implement 'v' is termed the virtul or effective hitch point; t the instnt shown, the implement behves s if it were ttched to the trctor t tht point. As n emple, Figure 6.5() shows plough on the three-point linkge s it enters the ground. It will be seen tht the effective hitch point is below the ground nd the line of drft psses bove it. The soil force hs clockwise moment bout tht point, thus the plough is being pulled into the ground. As this occurs, the effective hitch point rises nd eventully n equilibrium is reched where the downwrd force of soil on the plough is just blnced by the upwrd force of the trctor on the plough. The line of pull psses through the effective hitch point, now bove the ground surfce, s shown in Figure 6.5(b); this tends to dd weight to the rer wheels of the trctor. The bove is termed the 'free link' condition but it is not suitble for norml opertion becuse ny vrition in the direction of the soil force will cuse the implement depth to chnge. Usully, the linkge is rrnged so tht the implement reches the desired working depth before the effective hitch point rises up to the line of drft. The implement is thus kept from reching the equilibrium condition; the soil forces tend to pull the plough in deeper, but the linkge stops this occurring. The weight of the plough nd the downwrd cting soil forces re thus trnsferred to the rer wheels of the trctor. The line of drft psses bove the effective hitch point, s shown in Figure 6.5(c); the former cnnot be locted from the ltter s in the Figure 6.5(b). Further discussion is given in Dwyer (1974) nd Inns (1985). Problem 6.1 Tke mesurement of the three-point linkge system on trctor nd ssocited soil engging implement. Plot on drwing pper the position of the instntneous centre of rottion / virtul hitch point if the implement were rised nd lowered to below the ground level. Alter the linkge or use nother type of implement nd repet the bove. 6.3 TRACTOR CHASSIS MECHANICS The term 'mechnics' here refers to n nlysis of the forces tht ct on the trctor chssis. The mjor force is tht of grvity nd is known s the weight. This is sometimes (loosely) given, nd spoken of, in units of mss (kg); in engineering nlysis (concerned with sttics) ll such 'weights' should be converted to force units (kn) Centre of grvity The centre of grvity is the point t which the whole of the mss nd the weight of the trctor my be considered to ct. Its loction depends on the disposition of the vrious msses tht comprise the trctor. Any nlysis of the trctor chssis requires the loction of the centre of grvity to be known. It is usully specified in reltion to the rer le s shown by point G in Figure 6.6. () Longitudinl loction The loction of the centre of grvity in the longitudinl () direction my be found by mesuring the weight on the front (W f ) nd rer (W r ) wheels. Appliction of the force equilibrium condition gives the trctor weight, W: W = W f + W r Appliction of the moment equilibrium condition gives the required longitudinl loction, r s shown in Figure 6.6(). The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

7 6.6 β1 Wf W Wr f r () y' ' Wf' rf G G W β2 rr O 'r (b) " yg G' r yg G β z' 'r (c) r Figure 6.6: Loction of centre of grvity of trctor () horizontl loction (b) trctor rised to find verticl loction (c) geometry of position of centre of grvity Adpted from Brger, et l (1952) The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

8 6.7 For the trctor tke moments bout O: W. r = W f. r = W f W (6.1) The wheel bse () between the front nd rer les is usully given in the mnufcturer's specifiction or cn be mesured directly. For most common rer wheel drive trctors r is pproimtely 30 % of ; this is lso the % of the sttic trctor weight tht is on the front wheels. (b) Verticl loction The loction of the centre of grvity in the verticl (y) direction is more difficult. The common method is to lift the front (or rer) of the trctor (s shown in Figure 6.6(b)) nd mesure the weight on the front wheels (W' f ) in the rised condition. The following is similr to Brger et. l., (1952). Appliction of the moment equilibrium condition gives the required verticl loction, y g. For the trctor tke moments bout O: ' r = W' f W " (6.2) The geometry of the positions of the centre of grvity (Figure 6.1(c)) gives: z = 'r cosβ y g = r - z tnβ Substituting for z gives y g = ' r r - cosβ tnβ (6.3) where ' r is s clculted from Eqution 6.2 bove. nd β = β 1 + β 2 = tn r r - r f + tn y' - r r '' Inspection of Eqution 6.3 shows tht if the difference between r ' nd r ' needs to be reltively lrge nd / or ccurtely determined. cosβ is to be ccurtely clculted, β Problem 6.2 By similr mesurement nd nlysis to the bove find the loction in the verticl nd longitudinl directions of the centre of grvity of two wheeled trctor or triler. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

9 Issues in chssis mechnics Two spects of the mechnics of the trctor chssis, which re of importnce to the performnce of the trctor, cn be identified: () Weight trnsfer For trctor under dynmic (here mening 'operting') conditions, the weight on the wheels will, in generl, be different from the sttic vlues. These chnges re termed 'weight trnsfer' lthough of course nothing is 'trnsferred'. The discussion here is limited to the chnges in the v erticl longitudinl plne, ie, from front to rer nd vice vers becuse these hve the gretest influence on trctor performnce. Weight trnsfer is norml outcome of the ction of the forces generted on the trctor chssis by the ground nd by the implement. It occurs whenever nd however the trctor is loded, including the no lod cse where there is some weight trnsfer due to the torque on the rer wheels required to propel the trctor ginst the rolling resistnce of ll the wheels.. It is lso normlly desirble outcome becuse the trctor is designed to tke dvntge of it by hving t lest some of the driving wheels t the rer where, for norml forwrd opertion, the increse in rer wheel weight is proportionl to the drwbr pull. In reverse ger nd in the 'over-run' condition, (the implement pushing the trctor) the forces towrd the front of the trctor trnsfer weight from the rer wheels to the front wheels, fct which ffects the performnce of the trctor in this type of work nd when brking. A more detiled discussion of the generl subject of weight trnsfer is given in Gilfilln (1970), Liljedhl et l (1979) nd other references given t the end of this Chpter. (b) Instbility Instbility occurs when the weight trnsfer is sufficient to cuse the trctor to tip over rerwrds. I mpending instbility (where the front wheels leve the ground nd the trctor is on the point of becoming unstble) is considered here becuse it is limiting cse of the weight trnsfer nd hence of trctor opertion. It is n undesirble sitution becuse it represents loss of steering control nd my led on directly to ctul instbility. Such sitution is prtly voided by inherent fetures of the design of the trctor-implement system nd prtly by its opertion in wy tht voids reching tht condition. Usully the wheels slip before instbility occurs. An understnding of the ctul process of tipping over in the verticl longitudinl plne which my follow requires different, more comple dynmic nlysis tht includes, mong other mtters, the inerti of the trctor chssis nd of the implement, lso the inerti nd stiffness of the trnsmission to the rer wheels. This nd the nlysis of instbility in the lterl verticl plne (roll over) re not relevnt to trctor performnce s such; they re delt with in Liljedhl et l (1979) nd other references given t the end of this Chpter. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

10 Anlysis nd ssumptions The following nlysis of the trctor in the longitudinl, verticl plne is limited to the clcultion of wheel weight during stedy stte opertion in norml work (Section 6.4) nd to the prediction of the conditions for impending instbility (Section 6.5). Although the trctor nd implement re moving, the ssumption of stedy stte opertion implies tht there re no inerti forces; the forces re doing eternl work but re not cusing ny ccelertion. Hence the principles of sttics nd the conditions for sttic equilibrium of rigid bodies cn be pplied. Three independent equtions of equilibrium (chosen from the following) cn be written: (i) (ii) the sum of the forces in ny two perpendiculr directions re zero. The two directions usully chosen re those prllel to nd perpendiculr to the ground surfce. the sum of the moments bout ny two points in the verticl longitudinl plne re zero. The two points usully chosen re the wheel / ground contct points or the centres of the wheels. In simple situtions it my be sufficient to consider the whole trctor s rigid body. Where the eternl forces re known the weights on the wheels cn be clculted directly. However it is sometimes convenient to consider the trctor s composed of two rigid bodies. One, the drive wheels, rotte bout centre locted in the other - the chssis of the trctor. This occurs under the ction of the torque cting on them which is internlly produced by the engine. Any such nlysis must pply pproprite constrints ie, tht the forces nd moments on ech re equl nd opposite. In this nlysis nd the worked emples, the following simple ssumptions re mde: (i) (ii) (iii) (iv) forwrd motion is uniform; this ssumes constnt implement forces nd no ccelertion lines of forces on wheels re either tngentil or rdil or my be resolved s such; wheel sinkge nd tyre distortion (but not norml tyre deflection) re neglected the trctor is symmetricl bout the longitudinl verticl plne; ll the forces nd moments my be considered to ct in this plne other forces, such s the chnge in position of the fuel nd oil in the trctor on sloping ground, ir resistnce nd other minor forces re neglected The nlyses of trctors where other more comple ssumptions re mde re given in the references t the end of this Chpter. The trctor considered in the generl nlysis is s shown in Figure 6.7. The implement force P cts through the point (', y') t n ngle θ to the ground surfce. Note tht it is not shown ' ttched' to the chssis t the rer of the trctor becuse, in generl, it my ct on the trctor or ttched implement t ny point in the plne. For triled hitch shown in Figure 6.1, this point would be the drwbr / implement ttchment point. For the trctor in Problem 6.7, P is the weight of tnk nd wter ( verticl force) crried on the front. Cre must therefore be tken to ensure tht the direction nd the moment of P is correctly included by pproprite choice of θ nd the sign for '. The solution of the problems given in the following sections will be gretly fcilitted by coding of Equtions 6.4 nd 6.5, etc, on computer spred sheet. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

11 6.10 M Vf Q f f r W M C O Vr H θ P yg y y' α ' Figure 6.7: Trctor detils for weight trnsfer nlysis The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

12 WEIGHT TRANSFER Four wheel trctor () Anlysis Consider rer wheel drive trctor on slope s shown in Figure 6.7. For the trctor 1, tke moments bout C: V f + W sinα y g + P sinθ ' + M = W cosα r + P cosθ y V f = W cosα r + P cosθ y - M - Wsinα y g - P sinθ ' For the wheels, tke moments bout C: M = H. r Resolve prllel to the slope: H = W sinα + P cosθ Substitute for M nd H bove: V f = W cosα r + P cosθ y - W sinα r - P cosθ r - W sinα y g - P sinθ ' Combining: V f = W cosα r - W sinα r+y g - P cosθ r-y - P sinθ ' V f = W f - W sinα r+y g - P cosθ y' - P sinθ ' (6.4) Problem 6.3 Show tht the weight on the rer wheels (V r ) perpendiculr to the slope is given by: V r = W r + W sinα r+y g + P cosθ y' +' + P sinθ (6.5) 1 In the following, the totl weight of the trctor (W) nd the distnce to its centre of grvity ( r ) hve been used; this is stticlly equivlent to using the weight of the body (trctor less rer wheels) nd the distnce to its centre of grvity. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

13 6.12 (b) Eplntion of terms The terms in Equtions 6.4 nd 6.5 cn be identified s follows: (i) W f, W r the sttic weight on the wheels when the trctor is on the slope (ii) (iii) (iv) (v) W sinα r+y g P cosθ y' P sinθ ' P sinθ +' the moment effect of the weight component down the slope, decresing the front wheel weight nd incresing the rer. the moment effect of the implement force component down the slope, decresing the front wheel weight nd incresing the rer. the moment effect of the implement force component perpendiculr to the slope, decresing the front wheel weight. the direct (P sinθ) nd the moment effect (P sinθ ' ) of the implement force component perpendiculr to the slope, incresing the rer wheel weight. Referring to the Equtions 6.4 nd 6.5, note tht the moment effect of the component of the drwbr pull down the slope, P cosθ, hs two effects: (i) P cosθ y : increses V f nd decreses V r with moment rm y (ii) P cosθ r decreses V f nd increses V r with moment rm r The net effect of P cosθ is therefore the difference between these two, ie, P cosθ r-y y' = P cosθ. This fct gives rise to the ide tht if the drwbr pull cts below the rer le, its moment, Pcosθ. y, increses Vf nd holds the front of the trctor down. While this is true, it omits the more importnt, unrecognised spect tht usully lrger moment, Pcosθ. r, tends to decrese the weight on the front wheels. Problem 6.4 Check Equtions 6.4 nd 6.5 by tking moments bout the ground contct points O nd Q, respectively. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

14 6.13 (c) Specil cses The following specil cses re of interest: (i) If y' increses, ie, the point of ction (eg, the drwbr) is rised, y decreses nd the weight trnsfer, P cosθ r-y increses; the trctor my rech the condition of impending instbility when V f = 0 (Refer Section 6.5) (ii) If y' = 0, the point of ction (the drwbr) is t ground level, y = r; there is no weight trnsfer due to P. (iii) (iv) If y' is negtive, the point of ction is below ground level (eg, s is possible with three point linkge or y' with the drwbr in trench), y is greter thn r, the term P cosθ becomes positive in Eqution 6.4 nd negtive in Eqution 6.5, ie, weight is trnsferred from the rer to the front wheels. (Refer Section If θ = 0, ie, the implement force is prllel to the ground V f = W cosα r - W sinα r+y g - P y' V r = W cosα f + W sinα r+y g + P y' (v) If lso, α = 0, ie, the ground is horizontl V f = W r - Py' = W f - Py' V r = W f + Py' = W r + Py' (vi) If lso, P = 0, ie, there is no implement force V f = W r = W f V r = W f = W r Problem 6.5 Repet the nlysis in Section for the trctor trvelling down the slope where the implement force cts forwrds nd downwrds (s when towing n unblnced triler); show tht the wheel weights re: V f = W f + W sinα r+y g V r = W r - W sinα r+y g + P cosθ y' - P cosθ y' - P sinθ ' + P sinθ '+ (6.6) (6.7) The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

15 6.14 Problem 6.6 Consider the Frmlnd trctor with spry tnk mounted on the three-point linkge t the rer. The following dt pply: Weight of spry tnk when empty Centre of grvity of the tnk nd wter = 60 kg = 1.5 m from the rer le = 1.0 m from the ground (i) If there is 210 kg of wter in the tnk, wht is the weight on the front wheels for the unit moving on horizontl ground? (ii) Wht weight of wter cn be crried nd wht will be the trctive coefficient (bsed on the totl trctive force) if the unit is moving up 10 o slope nd the weight on the front wheels is to not be less thn 4kN? (iii) Wht will be the mimum weight on the front wheels nd the trctive coefficient s the trctor empties the spry tnk while trvelling down 10 o slope? Solution Prt (ii) From Eqution 6.4: V f = W f - W sinα r+y g - P cosθ y' ' - P sinθ P = W cosα r - W sinα (r+y g ) - V f cosθ y' + sinθ ' = 27.9 ( ) = 2.18 kn = 224 kg Weight of wter = = 164 kg From Eqution 6.5 V r = W r + W sinα r+y g + P cosθ y' +' + P sinθ = 27.9( = = 25.6 kn ) ( ) ψ' = W sinα + P cosθ Vr = = 0.20 Answers: (i) 5.92 kn; (ii) 164 kg, 0.20; (iii) 9.48kN, The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

16 6.15 Problem 6.7 Repet Problem 6.6 with the spry tnk mounted on the front of the trctor with its centre of grvity 1.5 m from the front le. (i) (ii) (iii) If there is 210 kg of wter, wht is the weight on the front wheels for the unit moving on horizontl ground? Wht weight of wter cn be crried nd wht will be the trctive coefficient (bsed on the totl trctive force) if the unit is moving up 10 o slope nd the front wheel weight is to not eceed 10 kn? Wht weight of wter cn be crried nd wht will be the trctive coefficient (bsed on the totl trctive force) if the unit is moving down 10 o slope nd the front wheel weight is to not eceed 14 kn? Answers: (i) 12.8 kn; (ii)187 kg, 0.26; (b) 165 kg; Problem 6.8 Consider the Frmlnd trctor operting up slope α = 15 o with drwbr pull ngle θ =10 o. Use Eqution 6.5 to clculte the : (i) mimum drwbr pull if the trctive coefficient ψ (bsed on the totl trctive force) = 0.8 (ii) rer wheel weight (iii) percentge contributions of the terms in Eqution 6.5 to the trctive force. Note: An itertive method is required to solve this problem becuse the rer wheel weight depends on the drwbr pull (due to weight trnsfer) nd the drwbr pull (s determined by the trctive coefficient) depends on the rer wheel weight. Assume n initil vlue for P nd clculte Vr, H nd then P; if the initil vlue of P is crefully chosen, the nswer will be obtined with sufficient ccurcy with two itertions. Answers: (i) 17.1 kn; (ii) 30.1 kn; (iii) 64%, 10%, 13%, 13% Weight trnsfer with rolling resistnce The bove nlysis neglects ny effect of rolling resistnce. We my, however, include this by introducing force cting long the slope (opposite the direction of motion) s further force to be overcome by the trctor. As discussed in Section the rolling resistnce my be epressed in terms of coefficient (ρ) s Rolling resistnce = ρ. Weight on wheel Here the weight will be the wheel weights perpendiculr to the slope, ie, V f nd V r s given by Equtions 6.4 nd 6.5 bove. The rolling resistnce for the trctor my be estimted by combining the effect on the front nd rer wheels by considering coefficient for the trctor s whole. The totl trctive force R = ρ (V f + V r ) = ρ (W cosα + P sin θ) H = W sinα + P cosθ + ρ (W cosα + P sin θ) The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

17 o o 5 o 5 o o o o 15 o -0.4 P=500 α=20 o -0.5 P=500 P=100 α=20 o -0.6 P= Trctive coefficient, up () up, ρ= up, ρ=0.025 down, ρ= down,ρ= Angle of slope, degree (b) Figure 6.8: Trctive coefficients required for the Frmlnd trctor working up nd down the slope : () crrying weight of 100, 300 nd 500kg with rolling resistnce coefficient of 0.05 (b) Crrying weight of 300kg with rolling resistnce coefficient of (bitumen rod)nd 0.1 (ploughed soil). The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

18 6.17 We cn specify the trctive force required (for rer wheel drive trctor) in terms of the gross trctive coefficient. ψ' = = Trctive force Rer wheel weight W sinα + P cosθ + ρ (W cosα + P sinθ) W r + W sinα r+y g + P cosθ y' + P sinθ +' (6.8) Problem 6.9 The Frmlnd trctor crries fertilizer distributor mounted on the rer three-point linkge. The following dt pply: Centre of grvity of distributor nd fertilizer, m: 1.5 behind trctor rer le 1.0 bove ground Totl weight of the distributor nd fertilizer, kg: 100 (empty), (full) Rolling resistnce coefficient (bitumen rod), (firm surfce) 0.1 (ploughed soil) Angle of slope (up nd down), o 0, 5, 10, 15, nd 20 Clculte the trction coefficient required to drive the trctor nd distributor under vrious conditions. Hence identify conditions where it my be possible nd sfe to drive up slope but unsfe to drive down it. Solution Results for some conditions which re given in Figure 6.8() for r=0.05 (firm conditions) show tht the trctive coefficient required: (i) increses with the ngle of slope (ii) decreses with weight crried, prticulrly for lrger ngles Figure 6.8(b) shows tht the trctive coefficient depends on the ngle of slope nd the rolling resistnce. In the emple given for lod =300 kg nd ρ = (bitumen rod), ψ'(down) > ψ' (up) for slope >12 o. Problem 6.10 Repet Problem 6.9 with the distributor mounted on the front of the trctor. Assume tht the centre of grvity of distributor nd fertilizer is 1.5 m in front of the front le nd 1.0 m bove the front wheel ground contct point. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

19 up down Trctive coefficient Figure 6.9: Slopes tht cn be negotited for vrious trction coefficients, Problem 6.11 The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

20 6.19 Problem 6.11 The Frmlnd trctor opertes with zero drwbr pull on slope α. The mimum gross trctive coefficient is ψ' nd the coefficient of rolling resistnce for the trctor s whole is ρ. (i) Wht is the mimum slope tht the trctor cn trvel up without eceeding the mimum trctive force. Resolving long the slope, Figure 6.7: H = W sinα + W cosα ρ At mimum gross trctive coefficient : H = V r. ψ' The dynmic weight V r on the rer wheels in opertion is given by moments bout Q: V r. = W cosα f + W sinα (r + y g ) ] V r = W cosα f + W sinα (r + y g ) Substitute for H nd V r bove: W sinα + W cosα ρ = [ W cosα f + W sinα (r + y g ) W sinα [1 - (r + y g ) f ψ'] = W cos α [ tn α u = ψ' f - ρ - ψ'(r+y g ) ψ' ρ] ] ψ' (6.9) (ii) Show tht the mimum slope tht the trctor cn trvel down without eceeding the mimum trctive force is: ψ' f + ρ tn α d = + ψ' (r+y g ) (6.10) (iii) Plot tn α u nd tn α d for vlues of ψ' between 0.2 nd 0.7 nd ρ = 0.05 nd discuss the mening of these results. Answer (iii) See Figure 6.9 The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

21 6.20 G C P Q W O S y' Vf V θ f r () G C P T Q W O S Vf V θ A y' z f r ' b (b) G C Q Vf W O V θ S y' z f r ' b (c) Figure 6.10: Weight trnsfer with vrious hitching systems; () triled; (b) semi-mounted; (c) fully mounted The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

22 Weight trnsfer with hitching systems () Anlysis Considering the three common hitching systems described in Section bove, we re now in position to evlute them with respect to weight trnsfer, ie, the increse in the weight on the rer wheels s result of the implement forces. This nlysis does not tke into ccount the weight of the implement, which is more significnt for the mounted nd semi-mounted systems thn for the triled. However, it provides vlid comprison of the reltive dvntges of weight trnsfer of the three systems on the bsis of the soil forces nd of the conditions under which these dvntges will be chieved. Consider n identicl cultivtor, s shown in Figure 6.10 hitched, in the following wys: (i) triled on its own wheels (ii) semi-mounted on the lower links of the trctor nd rer wheel (ii) fully mounted on the three-point linkge. In order to compre them it is necessry to determine the dynmic weight on the front nd rer wheels of the trctor for ech system; the sme soil force S, cting t n ngle θ to the ground surfce s shown, is ssumed for ech. (i) Triled Resolving horizontlly: P = S cosθ Moments bout Q for the trctor: V r = W f + P y' And V r V f = W r + = W f - S cosθ y' S cosθ y' (6.11) (6.12) Weight trnsfer will occur if V r > W r ie, if incresed by incresing the drwbr height, y'. y' is positive, ie, if the drwbr is bove ground level; it will be For considertion of the implictions of this, see the more generl nlysis of impending instbility given in Section 6.5. (ii) Semi-mounted Resolving horizontlly: P = S cosθ The dynmic weight T on the trctor drwbr is given by moments bout A for the cultivtor: T = S sinθ (-b) + P y' + S cosθ z where b gives the horizontl loction of the soil force. Substituting for P T = S sinθ -b + S cosθ z+y' The dynmic weight on the rer wheels is given by moments bout Q: V r = W f + P y' + T (+') The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

23 6.22 Vf < Wf Vr > Wr Figure 6.11 Conditions for weight trnsfer with fully mounted implement. HITCHING SYSTEM CONDITION FOR WEIGHT TRANSFER EXPLANATION TRAILED V r >W r unless y' negtive Drwbr bove ground level SEMI MOUNTED V r >W r lwys - - MOUNTED z V r >W r if tn θ > +'+b z V f <W f if tn θ > '+b Line of soil force psses bove:- front wheel/ ground contct point rer wheel / ground contct point Tble 6.1: Summry of conditions for weight trnsfer with vrious hitching systems The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

24 6.23 Substituting for T nd P: V r = W f + S cosθ y' + S sinθ (-b)(+') + S cosθ (z +y')(+') V r = W r + S sinθ (-b)(+') + S cosθ [ y' + (z +y')(+') ] = W r + S sinθ (-b)(+') + S cosθ y'+ (z +y')(+') (6.13) Problem 6.12 Show tht the weight on the front wheels of the trctor with semi-mounted implement is given by: V f = W f - S sinθ (-b) ' - S cosθ y'+(z + y')' (6.14) Weight trnsfer will occur if V r > W r which will lwys occur unless one of the following terms is negtive nd greter in mgnitude thn the other. The first term will be negtive if b >, ie, the soil force is behind the wheel. The second will be negtive if y' is negtive (below ground level) nd greter thn z or z is negtive (bove ground level) nd greter thn y'. All of these conditions re unlikely to occur for semi-mounted implement, hence weight trnsfer will lwys occur. (iii) Mounted The dynmic weight V r on the rer wheels is given by moments bout Q for the trctor / implement system s whole: V r + S cosθ z V r = W f + S sinθ (+'+b) = W r + S sinθ +'+b - S cosθ z (6.15) The dynmic weight V f on the front wheels is given by moments bout O for the trctor / implement system s whole: W r + S cosθ z V f = V f + S sinθ ('+b) = W f - S sinθ '+ b + S cosθ z (6.16) Incresing the length of mounted implements (hence incresing b) will increse the weight trnsfer to the rer wheels due to the direct effect (S sinθ) nd the moment effect (S sinθ '+b ) from the front wheels. The limit will be the length nd weight tht will still llow the trctor to lift the implement without itself tipping up; weights my be dded to the front of the trctor to void this. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

25 6.24 Weight trnsfer will occur if: V r > W r S sinθ +'+b > S cosθ z tnθ z > +'+b (6.17) This implies tht weight trnsfer to the rer wheels will occur if the soil force psses bove the front wheel / ground contct point, Figure The bove includes the contribution of the verticl component of the soil force (S sinθ) to the rer wheel weight. Another mesure ssocited with weight trnsfer from the front wheels in the mounted system is the condition tht Vf < Wf S sinθ '+b > S cosθ z z tnθ > '+b (6.18) This implies tht weight trnsfer from the front wheels to the rer will occur if the soil force psses bove the rer wheel / ground contct point. Further, weight trnsfer will increse s b increses, ie, the implement gets longer. (iv) Summry A summry of the results of this nlysis is given in Tble 6.1. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

26 6.25 semi-mounted > triled mounted > semi-mounted mounted > triled = -b -b Figure 6.12: Comprison of hitching systems on the bsis of weight trnsfer TOP GREATER THAN SIDE if line of soil force psses bove: MOUNTED SEMI MOUNTED Ais of lower hitch points: tn θ > z+y' b SEMI MOUNTED TRAILED Intersection, drwbr line & verticl line through front le: z+y' tn θ > +'+b Intersection, drwbr line & verticl line s fr in front of soil force s wheel is behind it: tn θ > z+y' -b Tble 6.2 Summry comprison of weight trnsfer effects for different hitching systems The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

27 6.26 (b) Comprison of hitching systems We seek to determine the conditions under which the weight trnsfer for ech system in Section () is greter thn the one bove it. (i) Condition for V r (mounted) greter thn V r (semi-mounted) : S sinθ +'+b sinθ [ +'+b - S cosθ z - (-b)(+') sinθ [b(++')] > S sinθ (-b)(+') ] > cosθ [ z > tnθ > + + y' + (z +y')(+') cosθ [(z +y')(++')] (z +y') b S cosθ [ y' + (z +y')(+') ] ] (6.19) For the weight trnsfer of the mounted implement to be greter thn tht for the semi-mounted, the soil force must pss bove the lower hitch points; see Figure (ii) Condition for V r (mounted) greter thn V r (triled) : S sinθ +'+b - S cosθ z > S cosθ y' sinθ +'+b z+ y' > cosθ z + y' tnθ > +'+b (6.20) For the weight trnsfer for the mounted implement to be greter thn tht for the triled, the soil force must pss bove the intersection of the drwbr line nd verticl line through the front le; see Figure (iii) Condition for V r (semi-mounted) greter thn V r (triled) : S sinθ (-b)(+') + y' + (z +y')(+') S cosθ [ ] > S cosθ y' sinθ (-b)(+') > cosθ [ y' tn θ > z+y' -b - y' + (z+y')(+') ] (6.21) For the weight trnsfer for the semi-mounted cultivtor to be greter thn tht for the triled, the soil force must pss bove the intersection of the drwbr line nd verticl line through point s fr forwrd of the soil force s the wheel of the semi-mounted cultivtor is behind it; see Figure (iv) Summry A summry of the results of this nlysis is given in Tble 6.2 The bove conditions re likely to be met with implements which hve; (i) soil force with significnt verticl component, such s mouldbord ploughs, compred to those with more horizontl force, such s cultivtors. (ii) long implements for which b is lrge. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

28 6.27 M U C M O θ yg y r W V ' H P y α h Figure 6.13 Two wheeled trctor dimensions relevnt to weight trnsfer nlysis The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

29 Other emples () Two wheel (wlking) trctor The wheels of two wheel ( or so-clled 'wlking') trctor re usully driven through V belt nd / or chin drive s shown in Figure 1.3. The mechnics of its chssis re the sme, in principle, s the conventionl fourwheel trctor but here the trctor chssis requires 'support' when pulling drwbr lod. This will usully be provided by one or more of the following: (i) tool, implement or triler t the rer (ii) wheel t the rer (iii) counter blnce weight t the front (iv) the opertor through the hndles Consider the two-wheel trctor s shown in Figure 6.13 on slope with n ngled pull through the drwbr. Normlly the loction of the centre of grvity would be such tht with no drwbr pull the trctor would tip forwrds nd countercting force U cting down on the hndles would be required. When drwbr pull cts the net moment on the chssis will be clockwise s in Figure 6.13 nd so the trctor tends to blnce itself. (i) With zero drwbr pull: For the trctor, tke moments bout O: U o h + W sinα (r + y g ) = W cosα r U o = W cosα r h - W sinα r + y g h (6.22) This force must ct downwrd s shown if the centre of grvity of the trctor, counter weight nd implement re forwrd of the le. (ii) With drwbr pull: For the trctor resolve prllel to the slope: H = P cosθ + W sinα Tke moments bout C for the wheels: M = H. r Moments bout C for the trctor: M + W sinα y g + P sinθ ' + U h = W cos α r + P cosθ (r-y') Substitute for H nd M from bove: P cosθ r + W sinα r + W sinα y g + P sinθ ' + U h = W cosα r + P cosθ (r-y') U = W cosα r h - W sinα r + y g h - P cosθ y' h - P sinθ ' h (6.23) = U o - P cosθ y' h - P sinθ ' h (6.24) The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

30 6.29 Problem 6.13 For the two-wheel trctor on slope nd with ngled drwbr pull, show tht the norml wheel weight is: V = W cosα h + r h - W sinα r+y g h - P cosθ y' h + P sinθ h ' h (6.25) For the convenient opertion of such trctor it would be desirble to rrnge tht the force U = 0 under operting conditions. Emintion of Eqution 6.23 (for simplicity with α = 0) shows tht this will depend on blncing the moment of the weight nd of the drwbr pull. W cosα r = P cosθ y' + P sinθ ' To chieve this it is common to ttch lrge weight t the front of the trctor, the position of which is djustble with respect to the le (equivlent to chnging r ) to chieve the desired blnce. r = P cosθ y' + P sinθ ' W cosα Problem 6.14 Show tht for the wlking trctor with α = 0 nd θ = 0, the condition for U=0 t mimum drwbr pull is tht r = ψ' y'. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

31 6.30 yt y G C y' A b W α () P D G T A W' R α (b) G M T A M W' R H α (c) Figure 6.14: Detils of PTO driven triler for nlysis The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

32 6.31 (b) PTO driven triler The PTO driven triler s shown in Figure 6.14 () cn be pulled up slope by trctor (6.14 (b)) or the wheels cn lso be driven vi drive shft from the PTO, (6.14(c)). (i) Pulled Consider the triler being pulled up the slope s in Figure 6.14(b). Resolve long the slope for the triler: P = W' sin α Moments bout D for the triler: R. = W' cos α b + W' sin α (y+y t ) R = W' cosα b - W' sinα y+y t Resolve perpendiculr to the slope for the triler: R + T = W' cosα Substitute for R: (ii) Driven T = W' cosα - W' cosα b + W' sinα y+y t = W' cosα -b + W' sin α y+y t Consider now the wheels being driven so tht the drwbr pull on the trctor is brought to zero s in Figure 6.14(c). Determine the trctive coefficient required for the triler wheels. Moments bout C for the triler: W' sinα y t + T. + M = W' cosα (-b) Moments bout C for the triler wheels: M Resolve long the slope: H = H. r' = W' sinα Resolve perpendiculr to the slope for the triler; R + T = W' cos α Substitute for T nd M bove W' sinα y t + (W' cos α - R) + W' sinα r = W' cosα (-b) R = W' sinα r + y t + W' cosα b ψ' = H R = W' sin α W' sin α r + y t + W' cos α b = tnα (r + y t ) tnα + b (6.26) The required trctive coefficient thus depends in comple wy on the slope ngle α nd the position of the wheels nd the centre of grvity. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

33 6.32 X G C D E Y Z G' Q W O G Vf f r Vr ' c b W' V' A () T G W' A (b) F P G' G T W' A V (c) Figure 6.15: Triled weight trnsfer hitch; () nd (b) without lift; (c) with lift The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

34 6.33 (c) Triled implement weight trnsfer system It ws shown bove tht the triled hitch is lest effective in terms of weight trnsfer. This deficiency hs been overcome by the development of weight trnsfer hitch, in which prt of the weight of triled implements, nd / or downwrd soil forces, re supported by the trctor rer wheels (Persson, 1967; Hockey, ). The principle of one common system is illustrted in Figure The rigid link XYZ is ttched to the threepoint linkge DY nd EY; Z is connected to the implement drwbr with fleible link ZG. In opertion, the three-point linkge pplies lifting force F to the implement; this is set by the opertor nd is kept constnt by hydrulic vlve even when the trctor pitches with respect to the implement. This support (but not lifting movement) of the implement trnsfers some implement weight, s well s some of the trctor front wheel weight, onto the rer wheels. Assume the weight trnsfer hitch is ttched to n unblnced triler s shown in Figure It is required to determine the weight on the rer wheels of the trctor when there is force F in the chin between the hitch nd the drwbr of the triler. Assume drwbr pull of P. (i) For the trctor nd triler with no lift nd no drwbr pull; Figure 6.15() nd (b). Moments bout A for the triler: T T = W' (-b) where T is the verticl force on the trctor drwbr = W' -b Moments bout O for the trctor: V f + T ' = W r V f = W r - T ' Substituting for T from bove = W f - T ' (6.27) V f = W f - W' (-b) ' (6.28) Moments bout Q for the trctor V r = W f + T ( + ') V r = W r + T + ' Substituting for T from bove: V r = W r + W' (-b)(+') = W r + W' -b + W' (-b) ' (6.29) The significnce of the terms in Equtions 6.28 nd 6.29 cn be identified s follows: W' -b = T - weight from the triler drwbr W' -b ' - weight from trctor front wheels due to T The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

35 6.34 (ii) For the trctor nd triler with pull P nd lift force F, Figure 6.15(c) Moments bout A for the triler W' (-b) + P y' + T = F (-c) where T is verticl force on the trctor drwbr nd is now ssumed to ct downwrds on the triler drwbr. T = F -c - W' -b - P y' Moments bout Q for the trctor V r + T ( + ') = W f + P y' + F (+'+c) V r = W r + P y' + F +'+c - T +' Substitute for T from bove: V r = W r + P y' + F +'+c - F (-c)(+'). + W' (-b)(+') y' (+'). + P. V r = W r + W' (-b)(+'). + P y'(++'). + F c(++' ). (6.30) Problem 6.15 Show tht the weight on the front wheels of the trctor with weight trnsfer hitch is: V f = W f - W' (-b)'. - P ('+)y'. - F c( + '). (6.31) The terms in these equtions showing the weight trnsferred to the rer trctor wheels cn be identified s follows: W r, W f, W' W' -b W' -b P y' P y' P y' F c ' - sttic weight on the respective wheels = T - weight on the triler drwbr ' - weight trnsferred from trctor front wheels due to T - weight from triler wheels due to P - weight from trctor front wheels due to P - weight from trctor front wheels due to trnsfer from triler wheels - weight from triler wheels due to F F c F c ' - weight from trctor front wheels due to F - weight from trctor front wheels due to trnsfer from triler wheels The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

36 6.35 M G C No contct W M O Vr H α P y yg y r ' Figure 6.16: Operting prmeters for trctor on slope with impending instbility The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

37 IMPENDING INSTABILITY The following nlysis, which is similr to tht given by Sck (1956), illustrtes the fctors which limit the opertion nd performnce of the trctor s result of impending instbility in the verticl longitudinl plne. Consider the four-wheel trctor on the slope with drwbr pull prllel to the ground surfce; θ = 0, s shown in Figure For impending instbility, ie, V f = 0 Moments bout C for the trctor: M + W sinα y g = W cosα r + P y Resolve perpendiculr to the slope: V r = W cosα Resolve prllel to the slope: H = P + W sinα Tke moments bove C for wheel: M = H r Write H = ψ' V r where ψ' is the gross trctive coefficient (ie, bsed on H ) Substitute for H, P nd M bove: M = ψ W cosα r Substitute for H, P nd M bove: ψ W cosα r = W cosα r + ψ W cosα y - W sinα y - W sinα y g sinα (y + y g ) = cosα ( r + ψ (y - r)) tnα (y + y g ) = r - ψ(r - y) tnα (r - y' + y g ) = r - ψ(r - y) ψ' = r - tnα (r - y' + y g ) y' ψ' = tnα + r - tnα (r + y g ) y' (6.32) Dividing through by (r+ y g ) gives ψ' = tnα + r - r + y tnα g y' r + y g The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

38 h= Trctive coefficient αs= o 30 o Angle of slope, o 40 o 0.0 o Tn(ngle of slope) αs=0.8 Figure 6.17: Reltionships for trctor with impending instbility The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln

39 6.38 Write r r + y g = tnα (sttic) = tnα s (6.33) α s is the ngle of slope tht would cuse the trctor to tip s rigid body bout the ground contct points under sttic, ie, no drwbr pull conditions; α s is usully lrge ngle, bout 40 o for most trctors. Write y' r + y g = drwbr height centre of grvity height = h (6.34) A typicl vlue for h is 0.6. ψ' = tnα + tn α s - tnα h (6.35) Here ψ is the trctive coefficient tht must be chieved to bring the trctor to impending instbility when it is operting on slope α. (i) (ii) (iii) If ψ' required to trvel up the slope is less thn ψ' given by Eqution 6.35, then the trctor will not rech impending instbility. If ψ' required to trvel up the slope is greter thn ψ' given by Eqution 6.35, then the trctor will rech impending instbility. If ψ' required to trvel up the slope is greter thn the mimum ψ' possible, then the trctor wheels will slip. Figure 6.17 shows plot of ψ versus tn α for vrious vlues of: (i) (ii) tnα s = 0.6 (high centre of grvity) nd 0.8(typicl centre of grvity) h = 0.6 (typicl drwbr height), 0.7 nd 0.8 ( high nd dngerous hitch point). The region where tnα > ψ' is not fesible; the trctor will slide off the slope. The emple shows trctor on the slope where tn α = 0.3. (i) (ii) For ψ' m = 0.8 (good trction conditions) instbility cn occur for h = 0.7 or 0.8 becuse ψ' m is greter thn ψ' = 0.72 or 0.67 required. For ψ' m = 0.6 (moderte trction conditions) instbility cnnot occur even for h = 0.8 becuse ψ' m is less thn ψ = 0.67 required; the wheels will slip. The generl conclusion to be drwn is tht impending instbility: (i) (ii) is unlikely to occur with norml drwbr heights, moderte slopes nd common trction conditions; usully the wheels slip my occur (often with ftl consequences) where trction conditions re good or hve been enhnced by the use of strkes (trction ids), where slopes re steep nd prticulrly where the drwbr or the loding point hs been rised. It should lso be noted tht, while the bove simple, sttic nlysis suggests the trctor is reltively sfe if used correctly, in prctice dynmic effects my influence its behviour nd crete dngerous situtions. For emple, ccelertion of the trctor forwrds introduces n inerti force through the centre of grvity tht hs moment bout the rer le which tends to tip the trctor rerwrds. The opposite will be true when the trctor is being brked; here weight is removed from the rer wheels which my dversely ffect their brking cpcity. The Mechnics of Trctor - Implement Performnce: Theory nd Worked Emples - R.H. Mcmilln