MATHEMATICAL MODELING OF THE TRACTOR-GRADER AGRICULTURAL SYSTEM CINEMATIC DURING LAND IMPROVING WORKS

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1 Bullein of he Transilvania Universiy of Braşov Series II: Foresry Wood Indusry Agriculural Food Engineering Vol. 5 (54) No MATHEMATICA MODEING OF THE TRACTOR-GRADER AGRICUTURA SYSTEM CINEMATIC DURING AND IMPROVING WORKS Gheorghe BRĂTUCU 1 Absrac: This paper sudies he influence of he pulled leveling machine on he general dynamics of he wheel racor. The obained daa are employed in he design of he ransmission and of oher racor sub-assemblies, represening he disurbance facors of he mahemaical and dynamic models se up of esablishing he srain on various componens. I is shown ha he rear axle of he racor is loaded up o 53%, while he fron one is released o values below he admissible minimum (20% of he saic load). The loading coefficien of he rear wheels can reach A 2 = 0.76, while he longiudinal sabiliy angle ils during climbing and worsens by abou 57% in relaion o he solo racor. Key words: agriculural racors on wheels, general dynamic, graders. 1. Symbols is Z 01, Z 02 - saic reacions on he fron and rear racor axels; Z 1, Z 2 - dynamic reacions on he fron and rear racor axels; G, G m - racor weigh, grader weigh respecively; F a, F j, F m, F f, F - forces of: air resisance; racor ineria; driving; rolling resisance; resisance on he working movemen of he echnical sysem; r m - dynamic radius of he driving wheel; M r1, M r2 - rolling resisance momens on he fron and rear racor wheels;, a, b, c, h c, h, h a - geomerical dimensions, according o Figure 1; f - rolling resisance coefficien; α, β - longiudinal and ransversal angles of he slopes; γ - angle made by he force F wih he horizonal; k u - soil leveling resisance coefficien; A - working surface (in soil) of he grader blade. 2. Inroducion Graders are working machines which cu he high irregulariies of he errain using blades or boomless buckes which ranspor rough creeping he released maerial o lower places [2]. From he echnical sysem formaion poin of view of he graders are owed machines, wih rigid or aricled frame (Figure 1). In Figure 1 we presened he echnical sysem scheme, made from U-650 M racor and he owed grader NT-2.25, a ramp ascension (a) and moving on ransversal slopes (b) [3]. 1 Dep. of Engineering and Managemen in Food and Tourism, Transilvania Universiy of Braşov.

2 130 Bullein of he Transilvania Universiy of Braşov Series II Vol. 5 (54) No Fig. 1. Movemen on longiudinal ramps (a) and ransversal slopes (b) of he racor-grader agriculural sysem The racion resisance opposed by he grader is calculaed wih he relaion: F = k A + G f. (1) u m Under he dynamic aspec, he grader influences he echnical sysem rough F forces magniude and he γ angle, on which forces are made wih an axel parallel wih he soil [1]. For his echnical sysem here will be considered in all he cases, he following consan values: racor weigh (G = N), racor wheel base ( = 2430 mm), graviy cener posiion (b = 850 mm, b = a = 1580 mm, h c = 900 mm) and racor gauge (B = 1920 mm). Variable values will be considered he ones which are relaed o he rolling resisance condiions and laeral slip, grader weigh, grader graviy cener posiion from he rear racor axel, racion resisance force and is posiion in relaion o he horizonal surface. The simulaion of he echnical sysem dynamics is made in MATHCAD language, using mahemaical models, corresponding o differen funcional siuaions and characerisic elemens of he general dynamics. 3. Mahemaical Modeling and Analysis of Tracor Axels Reacions As a reference elemen in analyzing he axels reacions of he racors which form he echnical sysems of land melioraion machines, are he saic reacions resuled from he racor weigh repariion on he wo axels, on horizonal errain: G b Z01 = = [ N], (2) G ( b) Z02 = = [ N]. (3) The dynamic reacions on he fron and rear wheels Z 1 and Z 2 are calculaed wih relaions: G b cos α ( G sin α + Fj ) hc Fa h a + F + F c g γ + M r1 + M r2 Z1 =, (4)

3 Brăucu, Gh.: Mahemaical Modeling of he Tracor-Grader Agriculural Sysem 131 Z 2 G ( b) cosα + ( G = M r1 + M r2 +. sin α + F j ) h c Fa ha + F + F ( + c) g γ + (5) The variaions of he dynamic reacions on hose wo racor axels are calculaed wih he relaions: Z. (6) 1 = Z1 Z01; Z 2 = Z 2 Z02 A he sudy of dynamic reacions i has been considered he case of a racor moving on a horizonal errain wih consan speed, wih he racion force parallel o he soil and neglecing air resisance. Resuls are presened in Table 1, from which we can find he influences of he main facors (F, γ and f) on he variaion of racor axels loads. Table 1 Reacion forces on he racor wheels a consan speed on a horizonal surface, wih he racion force parallel wih he soil surface Variables Z 01, Z 02, Z 1, Z 1 Z 2, Z 2 Symbols Values [%] [%] [N] [N] [%] [N] [N] [%] F, [N] γ, [ ] f Observaions γ = 30 f = 0.l F = f = 0.1 F = γ = 30 In he condiions of mainaining all he echnical sysem parameers consan, he resisan racion force from he grader loads on he rear axel of he racor wih 12% for F = 5000 N, wih 27% for F = N and wih 53% if F would have he value F = N. If racion resisance remains consan, bu he angle γ is modified, an angle which his force makes wih he horizonal, he load on he rear axel increases by 15.9% for γ = 0 and F = N, from F = 0, wih 27% for γ = 14 and by 42% for γ = 30. For he rolling resisance coefficien f = 0.06, F = N and γ = 30, he variaion of he loads on he racor axels owards he unloaded racor siuaion, shows a rear axle load increasing by 40% and a download of he fron axel by 74%. Increasing he value of f o f = 0.1 makes he rear axle load increase by 42% and he fron axel o discharge by 78%. In he hypoheical case where F = N, he rear axle load is of 47%, and he fron axel download of 87%. 4. Mahemaical Modeling and Analyses of oading Coefficiens for Tracor Wheels Tracor axels loading a he leveling work need o consider also he ire lif, expressed hrough wheel loading coefficiens λ. For he saic posiion of he racor hese coefficiens have he values given by he relaion: b λ fs = λ01 = = 0.35, (7)

4 132 Bullein of he Transilvania Universiy of Braşov Series II Vol. 5 (54) No b λ ss = λ02 = = (8) For he siuaion when he racor is moving wih racion force on a horizonal errain, he coefficiens λ f and λ s have he values: b F + f rm G λ f =, (9) G b F + f rm G λ s = +. (10) G The changing of hese coefficiens in he condiions of echnical sysem movemen on a plain errain, wih he variable force F of he grader, bu arranged in an angle of 30 from he horizonal and a rolling resisance coefficien f = 0.01, is represened in Figure 2. We found a decrease of he dynamic loading coefficien of he seering wheels λ f = λ 1 (curve 3) compared wih he coefficien λ fs = λ 01 (curve 1) of he same wheels, from he wheels λ f = λ 1 = from F = 5000 N, o he value λ f = λ 1 = for F = N. Fig. 2. The variaion of racor wheels loading coefficiens during leveling work A he same ime, an increase of he driving wheel loading coefficien (curve 4) from he value λ s = λ 2 = for F = 5000 N, o he value λ s = λ 2 = for F = N (curve 2 represens λ ss ) is noiced. 5. Mahemaical Modeling and Analyses of he ongiudinal Sabiliy of he Tracor For he deerminaion of he criical angles of saic sabiliy a boarding ramp (α 02max ) and ramp descending (α 01max ), he racor is considered slowed in boh siuaions and he following relaions are applied: b α02 max = arcg = h c b α01 max = arcg = h c o [ ], (11) o [ ]. (12) The criical angle of racor dynamic sabiliy a boarding ramp is calculaed wih he relaion: b ( ϕ f ) ( h + c g γ) gαcr =, (13) h ( h + c g γ) c if he angle γ has a significan value wih he relaion: b ( ϕ f ) h g αcr =, (14) hc h if angle γ = 0. The case of his agriculural sysem is no considered relevan in deermining he criical angles of longiudinal dynamic sabiliy a ramp descending, as a resul of his significanly improving wih he owed echnical sysem. In Table 2 he angle variaion of he owed echnical sysem longiudinal sabiliy is presened according o he resisance force F of he grader movemen. In he case of he racor-grader agriculural sysem s movemen wih a racion resisance F parallel wih he soil surface (γ = 0), he influence of he F force on back overhrow ( α 2 ) is in he limi of α 2 = 18.93%, if F = N and of α 2 = 9.2%, if F = 5000 N. However, if γ = 30, α 2 = 57.27%,

5 Brăucu, Gh.: Mahemaical Modeling of he Tracor-Grader Agriculural Sysem 133 The angle variaion of he owed echnical sysem longiudinal sabiliy according o he resisance forces F of he grader movemen Table 2 Variables α 01, α 02, α 1, α 1 α 2, α 2 Symbols Values [%] [%] [ ] [ ] [%] [ ] [ ] [%] F, [N] F, [N] F, [N] Observaions G = N h = 0.25 f = 0.1 γ = 15 G = N h = 0,25 f = 0.l γ = 30 G = N h = 0.25 f = 0.1 γ = 0 for F = N and α 2 = 16.8%, for F = 5000 N. More suggesive is he represenaion from Figure 3 of he γ angle influence over he longiudinal racor sliding sabiliy angles, in which 1 represens he posiion of longiudinal saic sabiliy criical angle a slopes descending (α 01max ); 2 - he posiion of he saic sabiliy criical angle a ramps ascending (α 02max ); 4 - he posiion of dynamic sabiliy angle a ramps ascending, if γ = 0, and 3 - he posiion of longiudinal dynamic sabiliy criical angle o sliding if γ = I is noed ha if he angle γ increases, he rear axel is geing loaded and he dynamic sliding sabiliy is improved. 6. The Mahemaical Modeling and Analyze of Tracor Transversal Sabiliy The saic sabiliy a agriculural sysem overhrow on ransversal slopes is calculaed wih he relaion: 0.5 B βmax = arcg. (15) h c For he maximum value of he racor U-650M rack widh (B = l.92 m) he saic sabiliy overhrow angle has he variaion represened in Figure 4, dependen only upon he racor graviy cener posiion h c. I is noiced ha for h c = 0.6 m, β max = 58, and for h c = l m, β max = 43, reaching a β max = 35, for h c = 1.4 m. The racor dynamic sabiliy a laeral overhrow shall be ensured if: G 2 0.5B cosβ hc sin β v R G, (16) g h c Fig. 3. The influence of angle γ over he saic and dynamic sabiliy angles a movemen on ramps and slopes in which he firs par (he cenrifugal force) is dependen on he speed v (for G and R consan on a racor), and he second par is dependen upon angle β of slope inclinaion.

6 134 Bullein of he Transilvania Universiy of Braşov Series II Vol. 5 (54) No Conclusions Fig. 4. The variaion of racor overhrow saic sabiliy angle depending on he posiion of he racor graviy cener h c Fig. 5. The variaion of dynamic sabiliy a laeral overhrow The condiion required by he relaion (13) is graphically represened in Figure 5, where on he abscissa, he movemen speed v and he slope angle β are represened, and on he ordinae, he cenrifugal force and he oher par of he inequaion are rendered. I is noiced ha he agriculural sysem s sabiliy is assured only if he movemen speed v < 2 m/s (curve 1), and he ransversal slope angle β < 30 (curve 2), a differen siuaion from he one sudied in he case of sabiliy a saic overhrow. 1. The moving resisance force F of he operaing racor-grader agriculural sysem modifies he loads on he racor axels by almos +27% on rear axel and by 49% an fron axel (a F = 15000N), he angle γ loads he rear axel by almos 26% and unloads he fron axel by 60%, and he rolling resisance coefficien causes he rear axel loading by a maximum of 7% and a fron axel unloading of 13%. 2. The fron axel unloading can no be acceped wih more han 80%, because he agriculural sysem s movemen direcions can no be safely conrolled. 3. The dynamic loading coefficiens of he racor wheels have similar variaions wih he axels loads, in he sense ha a he fron wheels (seering wheels) hose coefficiens decrease from 0.35 o ( 34%) and heir increase on he rear wheels from 0.65 o (+18.3%) because of force F and in smaller proporions if only he influences of γ and f are aken ino consideraion. References 1. Ionescu, E., e al.: Tracoare şi auomobile (Tracors and Auomobiles). Braşov. Transilvania Universiy Publishing House, Mihăilescu, S., e al.: Maşini de consrucţii (Building Machines). Vol. II. Bucureşi. Technical Publishing House, Năsăsoiu, S., Popescu, S., e al.: Tracoare (Tracors). Bucureşi. Didacic and Pedagogic Publishing House, 1983.