Agricultural and Biosystems Engineering Agricultural and Biosystems Engineering Publications 1967 Computer Simulation of an Unsprung Vehicle, Part I C. E. Goering University of Missouri Wesley F. Buchele Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/abe_eng_pubs Part of the Agriculture Commons, and the Bioresource and Agricultural Engineering Commons The ompc lete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ abe_eng_pubs/960. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. 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 Z23 = Z1 - Z11 + 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 Z4 = 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 Z2 = Z3 = 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) .