Dynamic effect of the bump steer in a wheeled tractor.doc

/locate/mechmtdynamic effect of the bump steer in a wheeled tractorp.a. simionescu a,*, d. beale b, i. talpasanu ca the university of tulsa, department of mechanical engineering, tulsa, ok 74104, usab auburn university, department of mechanical engineering, auburn, al 36849, usac wentworth institute of technology, electronics and mechanical department, boston, ma 02115, usareceived 8 june 2006; received in revised form 13 october 2006; accepted 21 october 2006 available online 1 december 2006abstractthe steering system of a compact wheeled tractor is analyzed using a commercial multi-body simulation software. the effect of axle-oscillation induced steering errors and of the axle impacting the bump stops attached to the tractor body is compared for two design variants. it is shown that by diminishing the kinematic cross-coupling between the steering-control linkage and the axle oscillation, a favorable reduction of the dynamic loads in the steering mechanism components can be obtained for normal operation conditions of the vehicle. however, if the axle impacts violently its bump stops, the reaction forces in the joints can be much higher; contrary to ones intuition, these forces are larger for steering systems with reduced or no bump steer. based on these conclusions, several design recommendations to the partitioning engineer are advanced at the end of the paper. 2006 elsevier ltd. all rights reserved.keywords: wheeled tractor; steering mechanism; rigid-body impact; multi-body software; joint forces1. introductionmost trucks and off-highway vehicles (like tractors, harvesting combines, self-propelled sprayers, fork lifts etc.), have rigid steering axles equipped with ackermann steering linkage. the input motion applied by the driver at the steering wheel is transmitted through the steering box and the steering control linkage (i.e. the pitman arm, drag link, steering arm assembly) to one of the steering knuckles normally the left side one (fig. 1).reports on the ackermann linkage analysis and design are available in a number of publications (see 1 and the references therein). this is a symmetric spatial four-bar function generator, which ensures that the steered wheels pivot at a certain ratio imposed by the condition of correct turning, viz. the ackermann law.the steering-control linkage has been also investigated by researchers, and some design recommendations can be found in the literature 15. the configuration schematized in fig. 2, where the axle is pin-jointed tocorresponding author. tel.: +1 918 6313198; fax: +1 913 6312397. e-mail address: (p.a. simionescu).0094-114x/s - see front matter 2006 elsevier ltd. all rights reserved. doi:10.1016/j.mechmachtheory.2006.10.007p.a. simionescu et al. / mechanism and machine theory 42 (2007) 135213611353fig. 1. ackermann steering linkage controlled via a drag-link spatial four-bar mechanism used in wheeled tractors and the like: (1) steering wheel; (2) steering box; (3) pitman arm; (4) drag link; (5) steering arm; (6) steering-knuckle arms; (7) track rod; (8) axle beam; (9) steered wheels.fig. 2. schematic of the ackermann linkage with rrssr (r stands from rotational joint, s stands from pivot joint) type steering control mechanism, where a the kingpin inclination angle and b the caster angles 1.the chassis, is specific to compact agricultural tractors 6. this is sometimes called live front axle suspension design 7, since it contains no deformable elements, apart from the wheel tires. conversely, trucks and some suvs have the axle attached to the chassis via elastic springs, and consequently the relative motion between axle and chassis includes both translations and rotations 810. paper 8 investigates the dynamics of the steering system of a heavy off-road prototype vehicle equipped with two steering axles, of which one proved to be insufficiently damped. papers 9,10 study the effect of braking forces, moments and weight transfer upon the steering-system compliances, together with the subsequent brake-steer and directional response of the vehicle.in the case of wheeled tractors and the like, however, there is no relative translation between the axle beam and the chassis, and consequently no brake-steer occurs. the steering-control linkage is a two degrees-of-freedom rrssr mechanism (without considering the redundant rotation of the drag link about its own axis fig. 2). these two dofs are: (a) the rotation of the pitman arm (the steering control motion) i.e. the change1354p.a. simionescu et al. / mechanism and machine theory 42 (2007) 13521361of angle u1 of the drop-arm measured from its reference position u01, and (b) the rotation of the axle relative to the tractor body by angle w measured from the horizontal. it is known to the automotive engineer that the cross-coupling between these two degrees-of-freedom should be as small as possible, because axle oscillations induce steering inputs which have a destabilizing effect upon the vehicle trajectory and can increased front tire wear 2,11. this cross-coupling can be defined as the added rotation, either of the steering arm for the pitman arm kept fix, or of the pitman arm for the kingpin joints locked, caused by the oscillation of the steer axle. in addition, as will be shown in this paper, if the axle impacts its bump-stops attached to the body of the vehicle, the reaction forces generated inside the steering-system components can increase significantly, to the level where they could induce static-load failure.2. multi-body modeling of the steering systemthe pro/mechanica motion package 12 was used for modeling the ackermann linkage and the rrssr control linkage of a compact wheeled tractor. the purpose of this undertaking was to validate the kinematic analysis results generated with an in-house computer program used previously 1,5, and to evaluate the dynamic effects of the cross-coupling between steering and axle oscillation, and of the occasional impacts that can occur between the axle-beam and the bump-stops mounted on the tractor body.same as in all parametric technology corporation products, in pro/mechanica the links of the mechanism to be simulated are parametrically defined, which allows for rapid modification and testing of different variants.fig. 3. pro/mechanica model of the steering mechanism. m and p are axle the impact zones and n and q are the bump stops attached to the tractor body.based on the dimensions extracted from manufactures drawings and real model measurements, the moving elements of the steering system of a compact tractor have been modeled using mass primitives (cylinders, cones, rectangular blocks and spheres see fig. 3). to add realism to the simulation, the tires have been modeled as massless toroidal surfaces attached to the wheel hubs. this feature of attaching more complicated geometries to the bodies can be useful in studying the working space of the moving elements of the mechanism, and in checking for unwanted interferences.p.a. simionescu et al. / mechanism and machine theory 42 (2007) 135213611355after modeling the individual links of the mechanism, they were connected to each other and to the tractor body using the following constraints (fig. 3): three pivot joints o1, o2 (the left and right kingpins) and o (the joint connecting the axle to the chassis); two pairs of spherical joints ab and cd connecting the track rod to steering-knuckle arms. (all pin joints and spherical joints were considered frictionless and with zero clearance); a zero dof joint (welded joint) a0 corresponding to a locked steering wheel. this is a common simplification 810, although in actuality there is some motion transmitted in reverse, i.e. from the pitman arm to the steering wheel, the amount of which depends on many factors like: the mechanical efficiency and gear ration of the steering box, moment of inertia the steering wheel and restraining torque applied by the driver at the steering wheel.3. load forces definitionthe motion of the axle and of the steering mechanism was studied relative to the vehicles body under the effect of the following forces: (a) gravitational forces; (b) a variable force f acting vertically upon the left wheel; (c) a pair of conditional forces that model the impact between the axle beam and its bump stops.by specifying the direction of the acceleration due to gravity, the software automatically applies the corresponding weights at the center of mass of each body in the present case gravity acts in the negative direction of the oz axis (fig. 3) with 9.81 m/s2 acceleration.the main external force considered for dynamic simulation was a single-period sinusoidal force f (fig. 4) applied vertically at the left wheel, which is known to be equivalent to a harmonic displacement input (see ref. 13, p. 239). this force is assumed to include the effect of tire stiffness (i.e. it is applied directly to the wheel rim), and could occur when the tractor traverses diagonally a bump or a ditch, or two slightly offset short bumps. the force can be also generated in laboratory conditions, with the tractor stationary and its front-end suspended, using a servo-controlled hydraulic cylinder that is jointed with one end to the ground and with the other end to the wheel hub. front axles of agricultural tractors can subject accidentally to much higher loads than the one considered here according to 14 a 35/55hp tractor was found to experience vertical forces at the front wheels in excess of 20,000 n.time sec0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4300 200 100 0 -100 -200-300fig. 4. time variation of the input force f applied at the left wheel.1356p.a. simionescu et al. / mechanism and machine theory 42 (2007) 13521361for the numerical example considered, the maximum oscillation angle of the axle beam is limited to 14 jj 14 by the bump stops attached to the body of the vehicle. in the simulation performed, these limits were imposed with the help of two conditional forces acting downward upon the axle beam at points m and p (see fig. 3). following 15,16 these two forces were defined as:fj=fs+fd(1)where fs is a elastic/repelling force and fd is a damping/dissipation forces (fig. 5), which act only when the distances mn or pq between the centers of impact become less than a certain small value sc. for the left-hand side impact (points m and n), the penetration of impact will be:( 0for mn sc oc mn for mn ocwhile for points p and q the penetration of impact will be:bright0for pq dcsc pq for pq sc(3)the spring force fs can be approximated using herz contact theory 15,16, and is a function of the physical and geometric properties of the two bodies, and of the penetration of impact. assuming a cylindrical axle beam of diameter d1, and each bump-stop being a semi-cylinder perpendicular to the axle beam of diameter d2 (fig. 5), the following equation holds 17:fd1d2t3/2/7 t7e1(1 v1) +2(1 v22) d1 + d21/2 s3?2(4)where e1, e2, m1 and m2 are the module of elasticity and poissons ratios of the materials in contact and k is a function of diameter ratio d1/d2 17. for the numerical example that will be further considered, e1 = e2 = 207 gpa, m1 = m2 = 0.3 while k = 0.723 (corresponding to axle beam diameter d1 = 80 mm and bump-stop diameter d1 = 16 mm).following 16,18, the damping force, which models the energy lost due to deformation of the two bodies, was set equal to:faks3/2s(5)3/2where k is the constant term that multiples d3/2 in eq. (4), while coefficient a depends on the material of the two bodies in contact. according to 16, a can be related to the kinematic coefficient of restitution e between the two bodies as:a = (1 e)/d0 with s0 the relative velocity between the two bodies just before the impact.(6)d2bump stop (body 2) -axle beam (body 1)ad11fig. 5. spring-damper model of the impact between the axle beam and the bump stops attached to the tractor body. all bodies are assumed ideally rigid, i.e. the effect of the vibrations generated during impact is neglected.p.a. simionescu et al. / mechanism and machine theory 42 (2007) 135213611357in summary, the two impact forces will be:rf 0for mn 8cf &5left (1 + 3/2 a 8cfrightk5!ht (1 + 3/2 a bright) for mtv j 8c(7)the left- and right-hand side penetrations of impact 8, and their derivatives 8 were defined in pro/mecha-nica as measures between points m-n and p-q. the threshold value 0 = 286, respectively. in turn, the parameters of the steering-control linkage were: steering-box output shaft location xa0 = 240 mm, ya0 = 734.5 mm, za0 = 315 mm, a0a = 127 mm, pitman arm reference angle p01 = 10, steering arm length, offset and reference angle b0b= 152 mm, 01b0= 148 mm and p02 = 193.the masses and moments of inertia of each moving link were automatically calculated by the software - all bodies were considered made of steel with a density of 7800 kg/m3. with these values and for the applied forces mentioned earlier, the time response of the system is shown in fig. 6, i.e. the axle oscillation jj(i) and left-wheel bump steer 9c(i), i.e. the added rotation of the left wheel due to axle oscillation for the pitman arm locked.fig. 7 shows a plot of the magnitude of the reaction forces occurring in the ball joints a and c of the steering control and of the ackermann linkage. both joints experience a steep load increase when the axle impacts1.414 12 1086420 -2 -4 -6 -8 -10 -12 -14fig. 6. time response of the existing steering system to the sinusoidal force f for the pitman arm locked in u = 0 position.1358p.a. simionescu et al. / mechanism and machine theory 42 (2007) 13521361time seca 1000100 10 10.10.010.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.21000100ijl 100.1time sec0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2fig. 7. time variation of the reaction forces in (a) ball joint a and (b) ball joint c of the initial design for the pitman arm locked in q = 0 position.the bump-stops. just before the impact, these forces are much smaller (15.9 n and 12.3 n, respectively), and are mostly caused by the inertia moment generated by the wheels about the kingpins as they are forced into a steering motion. the peak values of the same forces after impact are about 500 n and 300 n, which represent a 25-30 fold increase.the time response of the improved design is shown in fig. 8. the control linkage was redesigned by numerically searching for the steering-control linkage geometry that ensures minimum bump-steer 0c(t) as detailed in 1. the following geometric parameters had their values revised: a0a = 141.3 mm, 14 12 1086420 -2 -4 -6 -8 -10 -12 -14-lfig. 8. time response of the redesigned steering-system to the sinusoidal force f for the pitman arm locked in u = 0 position.p.a. simionescu et al. / mechanism and machine theory 42 (2007) 13521361135961 : existing design1 : improved design54321-14 -12 -10 -8 -6*-2 0 2 4 6 8 10 1214y deg0 -1 -2-3fig. 9. variation of the steering error dhc with the axle oscillation angle w for rectilinear motion.w 103 1000 1 1000.01time sec0.10.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 120.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2fig. 10. magnitude of the reaction forces occurring in (a) ball joint a and (b) ball joint c of the improved design. the 0.2 n increase of the reaction forces at ball-joint c is because the steering-