Agricultural and Biosystems Engineering Agricultural and Biosystems Engineering Publications

1967 Simulation of an Unsprung Vehicle, Part I C. E. Goering University of Missouri

Wesley F. Buchele Iowa State University

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This Article is brought to you for free and open access by the Agricultural and Biosystems Engineering at Iowa State University Digital Repository. It has been accepted for inclusion in Agricultural and Biosystems Engineering Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Computer Simulation of an Unsprung Vehicle, Part I

Abstract The mechanics of unsprung wheel tractors has received extensive study in the last 40 years. The quantitative approach to the problem essentially began with the work of McKibben (7) in the 1920s. Twenty years later, Worthington (12) analyzed the effect of pneumatic tires on tractor stabiIity. Later, Buchele (3) drew on land- locomotion theory to introduce soil variables into the equations for tractor stability. Differential equations were avoided in these analyses by assuming that the tractor moved with zero or constant acceleration. Thus, vibration and actual tipping of the tractor were beyond the scope of the analyses.

Disciplines Agriculture | Bioresource and Agricultural Engineering

Comments This article is published as Goering, C. E., and W. F. Buchele. "Computer Simulation of an Unsprung Vehicle, Part I." Transactions of the ASAE 10, no. 2 (1967): 272-276. DOI: 10.13031/2013.39652. Posted with permission.

This article is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/abe_eng_pubs/960 Computer Simulation of an Unsprung Vehicle Part I

G. E. Goering and W. F. Buchele M EMBER ASAE M EMBER ASAE THE mechanics of unsprung wheel Simplifying Assumptions As the chassis tips, the axles must tractors has received extensive study The following simplifying assump­ move on a fixed path relative to the in the last 40 years. The quantitative tions were made: center of gravity of the chassis. The approach to the problem essentially be­ 1 All motion of the tractor occurs in trigonometry illustrated in Fig. 2 gan with the work of McKibben (7)" a plane normal to the supporting sur­ yielded the required relationships, as in the 1920s. Twenty years later, Worth­ face. follows: ington (12) analyzed the effect of 2 The laws of Newtonian mechanics = - + pneumatic tires on tractor stab i Ii t y. are valid for measurements made on the H 11 sin (0 + 0 11 ) [1] Later, Buchele ( 3) drew on land-loco­ tractor relative to the earth. X23 = X1 + X11 + motion theory to introduce soil vari­ 3 Only the tractor tires are deforma­ H 11 cos (0 + 0 11 ) [2] ables into the equations for tractor sta­ ble; the remaining parts of the tractor = Z1 - ( Z11 + Z12) - bility. Differential equations were are rigid bodies. H12 cos (0 - 0 12 ) . • • [3] avoided in these analyses by assuming 4 The rear wheels have negligible X4 = X1 - (X12 - X11) + that the tractor moved with zero or mass but appreciable mass moments of H12 cos (0 - 0 12 ) . . . . • [4] constant acceleration. Thus, vibration inertia. Where the following relationships are and actual tipping of the tractor were 5 The front wheels have negligible implicit in equations [ 1] through [ 4] : beyond the scope of the analyses. mass and negligible mass moments of Z23 = Z1 + A Z23 • [5] Raney et al ( 10) established a mathe­ inertia. matical model for simulating the vibra­ 6 The front wheels are ground-driven X23 = X1 + A X23 . [6] tory behavior o{ a farm tractor. The only. Z4 = Z1 - A Z4 • • [7] differential equations in the model were 7 Damping in the pneumatic tires is X4 = X1 + AX4 • • [8] linear. Consequently, overturning and viscous damping. Differentiating equations [ 1] through large-amplitude vibrations were beyond 8 All wheel ballast is solid, not [ 4] , led to the following velocity equa­ the scope of the model. liquid. tions: _The primary objective of this study 9 There is negligible phase lag in w ~ s to establish a mathematical model the engine governor. Z23 = Z1 + H11 iJ cos (0 + 011 ) capable of predicting either large-am­ The fourth assumption can be real­ [9] plitude vibration or backward overturn­ ized only abstractly. Essentially all of ing of an unsprung wheel tractor. Other the mass of a rear wheel can be as­ X23 = x1 + H 11 iJ sin (0 + 011) objectives were to find a means for signed to the chassis. The small mass [10] solving the set of nonlinear differential remaining with the wheel can be im­ equations that constitute the model and agined to have a very large radius of [11] to obtain sufficient experimental data gyration, thus making possible an ap­ from a prototype tractor to check the preciable mass moment of inertia. ~4 = x1 - H 12 B sin ( 0 + 0d prediction accuracy of the mathemati­ ...... [12] cal model. This paper is 'confined to Notation The one acceleration equation required the work involved in meeting the first The dot notation was used to simplify in the model was obtained by differen­ two objectives. the writing of the equations. The dots tiating equation [ 10] : represent time derivatives, e.g.: DEVELOPMENT OF A MATHEMATICAL do ·~ 2 s = :;1 H 11 02 cos (0 + 0 11 ) MODEL () The mathematical model was based dt + H 11 0 sin (0 + 0 11 ) •••• [13] on: 0 The assumption of plane motion re- 1 Certain geometrical requirements, quired coincident motion of the rear 2 Newton's laws, and For the sake of brevity, the following wheels and allowed the following equa­ 3 Empirical information about sev­ derivations were made with the as­ tions to be written: eral components in the tractor. __ sumption that the reader would be fa­ = = Z23 . [14] Paper No. MC66-305 presented at the Mid­ miliar with the symbols. A complete Central Region Meeting of the American Society list of symbols is given in Appendix A. Z2 Z3 = Z23. [15] of Agricultural Engineers at St. Joseph, Mo., March, 1966, on a program arranged by the In general, lower-case symbols repre­ X2 X3 = X23 [16] Power and Machinery Division. Authorized for publication as journal paper No. 308_9 of th_e sent variables, while upper-case sym­ X2 X3 X23 [17] Missouri Agricultural Experiment Statton, proj­ bols are parameters. Fig. 1 is a draw­ = ect No. 515, and as journal paper No. 5483 of the Iowa Agricultural and Home Economics Ex­ ing showing the principal parameters X2 = X3 = X23 . [18] periment Station, project No. 1331. The authors-C. E. GOERING and W. F. on the tractor, as well as a three-dimen­ BUCHELE- are assistant professo r of agri­ sional coordinate system. This coordi­ Hitch Kinematics cultural engineering, University of Missouri, Col­ umbia, and professor of agricultural engineering, nate system, in which the z-axis is nor­ The line of pull of a drawbar load Iowa State University. mal to the supporting surface, is part varies with the angle of tip of the 0 Numbers in parentheses refer to the ap­ pended references. of a terminology which is being consid­ chassis. Thus a set of hitch kinematics Acknowledgment: The authors express -their appreciation of the loan of equipment, used in ered as an SAE recommended practice was necessary. In the following analy­ the study reported in this paper, by the product ( 2). The y-coordinate was not used in sis, the line of pull was assumed to pass engineering center of John Deere Waterloo Trac­ tor Works. the model developed in this study. through the center of resistance of a

272 TRANSACTIONS OF THE ASAE • 1967 ~- z

FIG. I Principal parameters and coordinate system used in the FIG. 2 Motion of front and rear axles relative to chassis center model. of gravity when the chassis tips positively. railed imp 1em en t, and tlu·ough the components in the tractor are shown in rear-axle center line yielded the fol­ hitch pin at the end of the tractor draw­ Fig. 4. Only one front wheel is shown, lowing equation for the chassis: ~ ar. In the hitch kinematics illustrated since insignificant differences were as­ I () in Fig. 3, the center of resistance is sumed to exist between the force sys­ shown above the surface. A center of () tems acting on the two front wheels. resistance below the surface requires a = 2F41 [X12 cos () + Z12 sin 8] Although the two rear wheels were kept negative Z 51 in the hitch kinematics - 2F42 [z12 cos () - x12 sin ()] equation. distinct, a free-body diagram is not + T2 + T3 - W1 [Xu cos (8 + ox) [ After reference to Fig. 3, the follow­ shown for the left rear wheel. Diagram W1 .. mg equations were written: b of Fig. 4 will serve that purpose if - Zn sin(()+ Ox)] + --Z1 g h15 = H13 sin v. . [19] the first subscript of every parameter and variable is changed to 3. Also, W1 v = () - a + 0 13 . [20] [ x11 cos () - z11 sin 8] + -- x1 = arcsin equations for wheel 3 will not be shown, g L 23 - z23 - Z51 - H 13 sin (8 + 0 13 ) since they are identical in form to those [ z11 cos()+ x11 sin ()] - p h10 for wheel 2...... [25] X 51 [21] Reversed effective forces and torques A summation of forces on the chassis here were applied on all of the free bodies in the z and x directions resulted in the following two equations: H 13 = \f'X213 + Z213 [22] to place them in dynamic equilibrium ( 8) . Reversed effective forces were not W1 .. 0" 13 - arctan - XZ13 . . [23] --z1 = -F21 - F31 - 2F41 13 shown on any of the wheels, which were g Equations [19] , [20] and [21 ] assumed to have zero mass. + W 1 cos Ox + p sin ( () + a) ere combined to yield the final hitch ...... [26] inematics equation: - p cos (()+a) [27] h15 = H 13 sin f () + 0 13 l A summation of forces on the wheels gave the following equations: L23 - Z23 - Z 51 - H 13 sin ( () + 0 13 ) ]) . - arcsin [ f ...... [24] ~1 ~1 [W] X 51 J F42 R42 [29] he pull inclination a and the moment ~l ~1 [~] h15 depend on both () and z23. The wheel rotations were chosen to F22 R23 R22 . [31] be positive as the tractor moved for­ The following equation was obtained ward. The positive sense of all other by summing moments on a rear wheel: variables is shown on the coordinate In this section, the forces on the trac­ system of Fig. 1. 12 c/> 2 = T2 - R23 Z21 + or will be related to the resulting mo­ R21 X21 + R22 ions. Free-body diagrams of the major A summation of moments about the .. [32] Pierrot ( 9) found that, for a given pneumatic tire rolling on a given sur­ face, the rolling resistance depends on the vertical load and forward velocity. In this study, the velocity effect was ignored. Rolling resistance can not ex­ ist, however, if the tire is not moving or tending to move forward. From these considerations, the following equations were written:

4 1 R12 = R41 • . • • • . • • • ___ I GROUND U N( /L4 [33] ----·------+---~- µ, 4 = 0 when ; 4 = 0 ... [33a] IG. '3 Hitch kinematics. R22 = /L2 R21 • • • . • • . . . [34]

967 • TRANSACTIONS OF THE ASAE 273 [38] is illustrated in Fig. 5 for a par­ ticular tire operating on concrete. The normal force between the sup­ porting medium and the tire depends on the motion of the tire normal to the surface. Each tire was visualized as a spring in parallel with a dash pot. The UJ ::> spring force is nonlinearly related to 0 a: positive radial deflections and is zero 0 when the tire is not in contact with the ..... surface. The damping force is assumed to vary linearly with the time rate of radial deflection of the tire. Although Raney et al ( 10) found the dynamic spring rate of a tire to be greater than the -statically measured spring rate, only FIG. 6 Torque-slip characteristic of typi­ the static force-deflection curve was cal clutch. used in the model developed in this Power Train Simulator study. Ground roughness contributes to the Summing torques on a hypothetical relative motion' across a tire. In the flywheel which contained all of the en­ model, the peaks and valleys in the sur­ gine inertia led to the following equa­ face were referenced from a datum tion: FIG. 4 Free-body diagrams of major com­ plane such that the average deviation ponents in the tractor. Tea = Te - l e ~ ...... [41] from the datum was zero. Upward ex­ The engine torque, Te, was taken di­ cursions in the surface were considered µ, 2 = 0 when ~ 2 = 0 ... [34a] rectly from the torque-speed curve of positive. The total radial deflection of the engine, i.e.: The thrust force of a rear wheel a front tire is given by (D4 + z4 +84 ) . varies directly with the normal force, Total deflections of the other tires are Te = T(w) ...... [42] i.e.: given by similar expressions. The use of relationship 42 in equation R2 a = /32 R21 ...... [35] The following equations express i:he [ 41 ] assumes negligible phase lag in the engine governor over the governed It is well known .that the slippage of relationships discussed above: range of the torque-speed characteristic. a powered wheel increases with the R - { f~ (D1 + Z4 + 84) + The clutch acts as a torque-limiting amount of thrust the wheel is required 41 - 0 device, as shown in Fig. 6. In this to deliver. The mathematical expression study, an idealized characteristic was of this phenomenon requires a relation­ C4(Z4 + ~4), (D4 + Z4 + 84) ~ 0 assumed as shown by the dashed lines ship between traction coefficient and , ( D 4 + Z4 + 84) ~ 0 in Fig. 6. This idealized clutch does travel reduction. The definition by ...... [39] not slip if the torque is less than Tm· If Barge et al ( 1) of travel reduction can the clutch is slipping, the torque must be expressed mathematically as: R _ { f2 (D2 + Z2 + 82) + 21 - 0 be T,w The mathematical expression of 20 2 TR2 = 27Tl - 27Tl • • • • • [ 36] the idealized clutch behavior is: 27Tl20 C2(Z2 + ;2), (D2 + Z2 + 82) ~ 0 Tc = Tea• w= wc ...... [43] where 120 , the no-load rolling radius, is ' (D2 + Z2 + 82 ) ~ 0 Ta, w> wc a constant for a given tire. By defini­ ...... [40 ] where the speed of the portion of the tion, the rotational and forward speed clutch connected to the transmission is: of a rear wheel are related through the In the preceding equations, f indicates a functional relationship on the quantity rolling radius Z2 as follows: W e = 2N ( 0 + ~ - ) . . . . [ 44] in parenthesis. However, C is a viscous 23 X2 = l2 ~2 ...... [37J damping coefficient and is multiplied and where· the average rear-wheel rota­ By multiplying numerator and denomi- by the quantity in parenthesis. tional speed is given by: nator of equation [36] by ~ , using 2 ~ - = 0.5 ( ~2 + ~3 equation [37] and simplifying, the fol­ 23 lowing equation for travel reduction was .. .. [45] obtained: The limiting torque is given by: Tm = }-te Te F e ...... [46] TR2 = 1 -~...... [36b] l20~2 For a given clutch, }-te and re are con­ TRACT IO N CH ARA CTER ISTI C stants ; md Tm varies only with the nor­ 15 5 11 38, 6 PLY TIRE Vandenberg and Reed ( 11) found an ON mal force. The normal force F e de­ empirical relationship between traction CONCRET E 18 PSIG, A'IR FILLED pends on the position of the clutch coefficient and travel reduction. For a STATIC LO AD• 4670 LB pedal, which is controlled by the oper­ powered wheel, the relationship can be ator. In starting the tractor from rest, expressed as: the operator usually engages the clutch /32 - µ,2 = g (TR2) . . . . . [38] slowly to provide a "smooth" start. He Vandenberg and Reed did not separate thereby slowly increases Tm from zero out the rolling resistance· of the pow­ to the maximum value at full engage­ ered wheel in their definition of traction TRAVE.L REDUCTIO N, •1. ment. coefficient. Thus the use of f3 2 - }-t2 in FIG. 5 Shape of functional relationship of The transmission system serves as a equation [38.J is correct. Relationship tractor coefficient to travel reduction. speed reducer, a torque multiplier, and,

274 TRANSACTIONS OF nm ASAE • 1967 ------

because of friction, as a power absorber. Since 0 was assumed to be small Through the use of equations [50], Input and output power to the trans- when the front wheels lifted: [64] and [65], equation [32] was mission are given by: · X12 cos 0 + Z12 sin 0 = X12 . . [60] simplified to: Input power = Tc W e ••••• [ 47] Z12 cos(} - X12 sin(} = Z12 . [61] 12 ~ 2 = T - Z21 (R2a - R22) Output power= T 2 (0 + 2 ) + For the same reason, equation [3] was [68] simplified to: Ta (0 + ~3 ) ...... [48] The Simplified Model Input and output power. are related as Z4 = Z1 - (H12 cos 012) 0 . . (62) In order to obtain a set of equations follows: Differentiating equation [62] led to suitable for programming, the preced­ (output power) = (input power) "f'J the following equations: ing equations were combined, simpli­ ...... [49] fied and manipulated as described be­ ~4 = ~t - (H12 cos 0 12 ). iJ . [63] low. The division of output power for a trac­ Worthington ( 12) found that, for a Equation [25] was multiplied tor with an unlocked differential is such rear tractor tire operating on a rigid through by dt and equations [28] , that equal torque is delivered to the surface, the normal support force is al­ [29] , (33] , [50] , [58], [60], and rear wheels, i.e.: most directly below the wheel center. [61] were used to obtain the follow­ T2 = T 3 = T ...... [50] ing moment equation for the chassis: By using equations [ 45] and [50]: equation [ 48] was simplified to: I do = 2(X12 - J.L4Z12) R41dt + 2Tdt - (W\X11) cos (0 + ax)dt (} Output power = 2T (0 + ~ -) 1 11 23 + (W1Z11 ) sin (0 + ax)dt + j W X ~ z.1 (cos (})dt l g ) t Wh~~ e~u~t:o~s· [~~] ·, ·[~7·] : [49][!~~ ·r w z 1 .. r w iZ11 1 .. [51] were combined and simplified, - ~-2...... !..!..I z1 (sin (})dt + ~---/x 1 (cos O)dt the result was: l g J l g J T = N TJ Tc ...... [ 52] + JW1X11 } ~~ (sin O) dt - (H13 ) p sin((}+ 01a)dt .... [69] Equations [44] , [45] , [50] and [52] \ g descripe a transmission-final drive sys­ The slope angle ax was retained for tem with an unlocked differential. '.fherefore, for a tractor operating on generality. For the assumed level con­ Equations [53] through [57] , not concrete: crete strip, ax = 0. shown, were derived by Goering [6] X21 = 0 ...... [64] Equation [26] was multiplied for a transmission-final-drive system However, the above approximation be­ with a locked differential. through by gdt and equations [28] , comes poorer as the drawbar pull in­ W1 APPLICATION TO BACKWARD TIPPING creases. On a level concrete surface, [30] , [59] and [66] were used to the thrust and rolling resistance forces The model was used to predict the obtain: behavior of a prototype tractor under two conditions. The first was sudden dz1 = - 2 {__L}R21 dt - 2f__L} R41 dt + (g cos aJdt engagement of the clutch of a tractor W1 tW1 without a drawbar load. The second + { ; } p (sin O)dt ...... [70] was slower engagement of the clutch 1 of a tractor pulling a heavy drawbar load. In both cases, the tractor was Similarly, equation [27] was multiplied started from rest and the supporting act ·at the tire concrete interface. There­ through by gdt/ W 1 and equations surface was level concrete. The proto­ fore: type was a heavily ballasted John Deere Z22 = Z21 = Lza · [65] [29], [31], [33], [59], and [67] 3010 tractor with an unlocked differ­ F21 = Fa1 · · · · · [66] were used to obtain: ential. The two rear wheels were identi­ cal and had equal i~Hation pressures. d;1 = 2 {~} (R23 - R22 ) dt - 2 fg µ,4} R41 dt When the prototype pulled a drawbar W1 l W1 load, the hitch was "long" and the line of pull was initially level. - (g sin ax) dt - _!L p (sin 0) dt ...... [71] The following two simplifying as­ W1 sumptions seemed appropriate for this Equations [63] and [39] were used problem: Since the two rear wheels were iden­ in determining the support force on a (a) The angle of tip, (}, is "small" tical and operated on the same surface, front wheel. These equations were mul­ when the front wheels leave the sup­ they were assumed to behave identi­ tiplied through by dt, yielding, respec­ porting surface. cally. Therefore: tively: (b) The requirement that µ 4 = 0 [72] when ; 4 = 0 can be ignored. Simplifying Equations Because of the "long" hitch, a was [73] assumed to be zero for all values of 0 (Fig. 3) and equation [24] simpli­ Fz2 = Fa2 ...... [67] Equations [74] and [75] were nec­ fied to: For the same reason, only one moment essary in determining the support force h15 = H13 sin (0 + @13) ... [58] equation for a rear wheel, equation on a rear wheel. Equation [74] was Also, (0 + a) = 0 ...... [59] [32], .was needed for the problem. obtained from a combination of equa-

1961 • TRANSACTIONS OF THE ASAE 275 tions [9] and [15]. Equation [75] APPENDIX A: NOTATION DI> static tire deflection of the it follows from equation [ 40]. The re­ Body 1, b·actor chassis wheel sulting equations were: Ci> viscous damping constant fo Body 2, right rear wheel wheel i dz2 = ~ dt Body 3, left rear wheel 1 f i> ( ) , functional load-deflection re­ ) ())d() + (H11 cos 0 11 (cos Body 4, front wheel lationship for wheel i (H11 sin 0 11 ) (sin 8)d() Body 5; supporting surface f 10, static support force on the itl> ...... [74] wheel I (f2(D2 + z2 + s2)dt + (dz2 + dsz) C2, (D2 + Z2 + s2) """ 0 gi> ( ) , functional traction-slip rela­ R21dt = r O tionship for wheel i J , (D2 Z2 Sz) ~ 0 + + TRI> travel reduction of the ith wheel [75] F1i, the jth force acting at the axle 0£ wheel i In equations [73] and [75], the sur­ R, the ;th supporting surface reactio (), angle of rotation of chassis about face-roughness variables were obtained action on wheel i an axis parallel to rear-axle cen­ for generality. For the assumed level p, the drawbar pull acting on th terline concrete surface, s4 = s2 = 0. tractor cf>i> angle of rotation of wheel (i) Equations [I3] and [18] were used W i, the weight of the chassis, whic xi> displacement of center of gravity was assumed equal to the weigh to obtain the differential d~ for later 2 of body (i) in the x-direction of the tractor use in the problem. The result was: zi> displacement of center of gravity h the centroidal mass moment of of body ( i) in the z-direction d~2 = d~1 + H11 cos (8 + 0 11 ) 02dt inertia of wheel i x23, displacement in the x-direction I () , the centroidal mass moment of + H 11 sin (8 + 0 11 ) dO . .. [76] of the intersection of rear axle cen­ inertia of the chassis about the Equations [77] , [38], [78] and terline with chassis centerline pitch axis [79] below were required to generate z23, displacement in the z-direction Ie, the effective mass moment of in­ of the intersection of the rear-axle ertia of the entire engine and (R - R )dt. Equation [77] d~2 23 22 centerline with the chassis center­ w, engine rotational speed follows directly from equation [68]. line Te, the engine torque corresponding Equation [78] was obtained by differ­ , h15 a variable moment arm defined to the speed w entiating equation [ 36b] and equation in Fig. 3 Ten• the sum of engine torque and [79] was derived from a combination a, inclination (in the x-z plane) of engine inertia torque of equations [34] and [35] . The re­ the drawbar pull Tm' the torque in clutch when it is sults were: , magnitude of when () = 0 a 0 a slipping v, certain angle defined on Fig. 3 1 d¢ = T dt - Z (R - R ) dt Ti> the torque in rear axle of wheel i 2 2 21 23 22 si effective surface roughness "seen" ...... [77] Tc, the torque in the clutch at anyj by wheel (i). When s1 = 0, the time (/32 - µ,2 ) = g (TR2 ) • • • [38] ground line is the datum N, the ratio of T1 to Tc li> effective rolling radius of wheel i 'Y) , the mechanical efficiency of the 1 d X2 zio> effective rolling radius of wheel i d(TR2 ) = - - I- 1 [78] transmission and final drives when drawbar pull is zero Z20 1~ 2 I F c• the normal force acting on the 0 1J, the jth fixed angle on body i friction facing of the clutch (R23 - R22) dt = xii> the jth horizontal dimension on /1-c• the kinetic coefficient of friction (/32 - P,2) R21 dt ...... [79] body i of the clutch facings The equations for the tipping prob- zii• the jth vertical dimension on re, the effective radius of the clutch lem are listed in Table 1, along with body i facings. the variables and parameters. Deriva­ H1i, the jth fixed hypotenuse on body tives of variable were not counted in i, fully defined in the text References , 1 Barger, E. C., Liljehdahl, J. B., Carleton, the list of variables. Note that the num­ L23 height above datum of the in­ W . M., and McKibben, E. G. Tractors and theu ber of variables equals the number of tersection of rear-axle centerline power units (2nd edition) New York N.Y., John Wiley and Sons, Inc., 1953. equations; thus the model is determinate with chassis centerline 2 Bidwell, J. B. Vehicle directional . control for the backward tipping problem. The 8x, slope of datum in the x-direction described in more precise terms. Society of Automotive -Engineers Journal 72:26-33, 1964. drawbar pull and the limiting clutch µ, 1, coefficient of rolling resistance of 3 Buchele, W. F. Mechanics of a vehicl torque Tm must be known as functions wheel i operating on a yielding soil. Pal'er presente.d at the Society of Automotive Engmeers, National of time, however. /3i> coefficient of tractor of wheel i Farm, Con·struction and Industrial Machinery Meeting, Milwaukee, Wis. September 10~13, 1962, (Mimeographed ) New York, N.Y., Society TABLE 1. LISTS OF EQUATIONS, VARIABLES, AND PARAMETERS of Automotive Engineers, Inc., 1962. FOR THE BACKWARD TIPPING PROBLEM 4 Farris, G. J., A digital differential analyzer programming system for the IBM 7074 com­ puter. Unpublished PhD thesis, Ames, Iowa. Equations Variables Parameters Equations Variables Parameters Library, Iowa State University, 1964. 5 Farris, G. J. and Burkhart, L. E. A pro­ 69 () I() 79 c/>2 D4, C4 gramming guide for DIAN. Unpublished report. Ames, Iowa. Ames Laboratory, U.S. Atomic En­ 70 R41 X12 41 H11, 011 ergy Commission, Iowa State University, 1964. /32 - P,2 6 Goering, C . E. Mechanics of unsprung 71 T /J-4 42 TR2 D2, C2 wheel tractors. Unpublished PhD thesis, Ames, Iowa, Library, Iowa State University, 1965. 72 Z1 Z12 43 Ten I2 7 McKibben, E. G. Kinematics and dynamics 73 x 1 W1 44 Te Z21 of the wheel-type farm tractor Agricultural En­ gineering 8: 15-16, 39-40, 58-60, 90-93, 119- 74 p Xn 52 w Z20 122, 155-160, 187-189, 1927. 75 p = p(t) 8 Meri am, J. L. Mechanics, Part II: Dy­ R21 8x Tc le namics. New York, N.Y., John Wiley and Sons, 76 R23 - R22 Zn Tm = T(t) Tm N,'YJ Inc., 1955. . 9 Pierrot, V. C. III. A similitude study of an 77 Z4 g unpowered pneumatic tire. Unpublished M.S. 38 Z2 H13, 0 13 thesis, Ames, Iowa, Library, Iowa State Uni­ versity, 1964. 78 X2 H12, 012 (Continued on page 280)

276 TRANSACTIONS OF THE ASAE • 1967