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Stability and control of two-wheel drive tractors and machinery on sloping ground
J. ugric. Engng Res. (1978) 23, 169-188
Stability and Control of Two-wheel Drive Tractors and Machinery on Sloping Ground H. B. SPENCER* This paper examines conditions under which the overturning stability and directional control of two-wheeled drive tractors with mounted or towed implements are lost on sloping ground. Theoretical methods are derived for predicting these conditions. Confirmation of the predictions by experiments on full size machines and models was obtained. The theoretical methods are used to examine the suitability of some frequently used tractor/ implement combinations for use on sloping ground. It is concluded that loss of wheel/ground adhesion, rather than the likelihood of overturning, limits the slope on which most combinations can safely operate.
1. Introduction The stability of a tractor against overturning on sloping ground has received much attention. Overturning does occur on such land but it is unlikely that lack of inherent stability is the primary cause of accidents. It is the author’s contention that accidents on sloping ground are likely to originate from loss of control of the tractor/implement combination followed by an increase in speed. On sloping ground rear wheel adhesion is likely to be lost, especially during descent, and, with retention of front wheel steering control, a dynamically unstable situation arises. If the machine slews an overturn due to centrifugal force can occur due to a high rate of turn.’ This contention that overturning stems in the first instance from control loss is based on an examination of the stability and control of machines operating on sloping ground. The examination, described here, was carried out by the construction of polar diagrams. Polar diagrams were used by Reichmann and Daskalov to show combinations of slope angle, p, and heading angle, a (Fig. 1) at which instability sets in. *s3v* The diagrams have hitherto been confined to vehicles not acted on by external loads and to stability loss, which occurs when the normal to ground reaction of one of the machine’s wheels becomes zero. This paper extends the diagrams to cover instability resulting from external loading and also to take account of wheel/ ground adhesion being inadequate to withstand braking, side or traction forces. Inadequacy results in the driver being unable to drive the tractor on an intended path and is referred to as a control loss. The problems examined are restricted to those arising in steady state straight line motion. Control loss is dependent on the ground conditions. In the absence of sufficient data to provide the general traction/side force/braking characteristic of tyres the work described here is based on friction concepts. This approach is not unduly restrictive since serious control loss problems occur with hard ground conditions for which friction concepts apply.5 The mathematical model describing the control loss boundary, i.e. the critical combinations of slope and heading angles, can easily be extended to cover soft ground conditions once wheel behaviour under these conditions can be described. One feature of the polar diagram depicting the stability and control loss boundaries of a machine is that, in the limit of infinite coefficient of friction, the two are identicalThis is readily seen if one considers that under these conditions control will only be lost when the wheel normal to ground load becomes zero, which corresponds to the stability limit. Extensive use is made of vector algebra to determine expressions for the forces acting on the wheels.* For a vehicle, this technique allows simple formulation of the equilibrium equations for Institute of Agricultural Engineering, Bush Estate, Penicuik, Midlothian, Scotland ‘Scottish Received 16 September 1977; accepted in revised form 8 January 1978
169 OOZI-8634/78/0601~169
$02.00/O
Q 1978 The British Society for Research in Agricultural Engineering
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Fig. 1. Definition of heading angle a and slope angle /I
any geometric configuration, such as a conventional two-wheel drive tractor with a pivoted front axle beam. The equations when solved for the wheel forces, from which the prediction of stability and control loss are made, give rise to lengthy expressions which are evaluated by a computer program. Theoretical predictions of control loss are confirmed by limited tests on two full size tractors and by model experiments on a variety of tractor configurations. The agreement is good enough to warrant use of the techniques for examining circumstances under which control of some commonly used tractor/implement combinations is lost. 2. Prediction of stability and control loss The prediction of combinations of slope /I and heading angle a at which a control or stability loss occurs requires the consideration of the force and moment equilibrium of the tractorimplement combination. This enables wheel ground reactions to be evaluated from which predictions can be made of the machines stability and control loss boundaries. The methods used in determining the wheel ground reactions and criteria for the occurrence of a stability or control loss are given in detail in Appendix A. A general configuration of a two-wheel drive tractor with trailed implement is considered in Fig. 2. The prediction techniques make use of a computer to evaluate the reactions at a fixed heading angle. The slope /3 is increased from zero by small steps until a stability or control loss occurs. By varying the heading angle from 0” to 360” complete stability and control loss diagrams can be determined. 3. Experimental results Experimental confirmation of predicted safety limits on machines poses investigators with problems. Tests on full scale machines to determine limits of safe performance on sloping ground
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KEY TO SYMBOLS A B C D BT CT Qr br Cl d,
denotes tractor wheels
denotes trailer wheels i vector position of wheel/ground
contact points of tractor, m
vector position of wheel/ground
contact points of trailer, m
I
:;: Pt
I-
Ptl Ai Br
vector position of drawbar hitch point on tractor, m vector position of drawbar hitch point on trailer, m vector component
Ci D, CT, Pk
of tractor wheel forces acting at wheel/ground
contact points, kg
1
J
vector component of trailer wheel forces acting at wheel/ground contact points, kg vector component of drawbar hitch load, kg component of unit vector describing the hinge axis of the front axle, m component of a position vector defining a point on the front axle hinge axis, m
;:
weight of front axle inclusive of ballast carried by axle or wheels, kg weight of tractor aft of front axle beam pivot, kg component of the vector giving the position of the centre of gravity of the front axle, m
Sl
; P & i,i,
G, CT,
6,
R
k
component of the vector giving the position of the centre of gravity of tractor excluding front axle, m component of the vector giving the position of the centre of gravity of the complete tractor, m heading angle, degrees slope of land (% slope = 100 x tan /?), % coefficient of sliding friction permutation symbol subscripts taking values 1, 2, 3 vector component of gravity force for tractor, kg vector component of gravity force for trailer, kg vector component of gravity force for front axle, kg rolling resistance, kg
have a high probability of severe accidents occurring involving irreparable damage to expensive machinery or endangering life and limb of operators. Previous investigators have used only one full scale machine in their trials.6 Some investigators have restricted testing to small scale models in order to obtain confirmation of their mathematical models.‘* * The approach used here is to use limited full size trials supplemented by a wide range of tests with small scale models. 3.1. Experiments on full scale tractors By their nature stability and control loss boundaries are not easy to determine
experimentally
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Tractor co-ordinate
AND
CONTROL
ON
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GROUND
system
k7;
Troller co-ordinate
system
Fig. 2. Co-ordinate system for conventional tractor, with hinged front axle, and trailer
for full size machines. Full size manually controlled experiments to determine control loss can only be conducted in safety for the direct descent case. The requirements are that the slope should not be very long and that a long flat run out area be available at the bottom. Trials were made with two different medium horse power tractors, X and Y, to determine the gradient on which control was lost during direct descent of hard grass slopes. One of the tractors was in the unballasted condition (X). The other (Y) was abnormally ballasted so that control was lost on a relatively moderate slope. Dimensions of tractors X and Y used in these trials are given in Appendix B. The tractors were driven down successively steeper slopes until control was lost. Engine braking, not wheel braking, was used and when wheel/ground adhesion was insufficient for equilibrium the tractor slid bodily downhill with the drive wheels rotating. The suddenness of the loss of control was a feature of these failures. The driver reported that no warning signs were available, such as noticeable wheel slip. This was confirmed from films taken during the trials. It would therefore appear that under hard ground conditions the increase in braking wheelslip attributable to increase in slope under constant engine braking is so small as to be not noticeable by the driver until wheel ground adhesion is insufficient for equilibrium and control is lost. This observation, no noticeable slip, is confirmed by results obtained from testing automobile tyres on hard surfaces.* In the idealized case of a pure friction model with a rigid wheel on hard ground there is no slip at the wheel/ground contact point until the friction force necessary for equilibrium exceeds the maximum available. The dangers inherent in this wheel behaviour are obvious.
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-Fig. 3. Theoretical control and stability loss diagrams for tractor X (lef) (, control loss: stability) and tractor Y (right) (- - - - -, control loss; - -, stability). Experimentally determined control, loss points at a = 180”; ?? , tractor X; x , tractor Y
3.2. Polar diagram representation of stability and control loss The computed stability and control loss boundaries for the two tractors, X and Y, are shown in Fig. 3 together with the experimentally determined control loss points for direct descent. This information is portrayed on a polar diagram with a0 as the circumferential and /I% as the radial co-ordinate. The use of /3 expressed as a ‘A (8% = 100 x tar@) in the polar stability diagram follows Reichmann who showed that the stability boundaries consisted of straight lines in this co-ordinate system which obviously eases construction of such diagrams.2 No such pronouncement on the shape of the control loss boundaries is yet possible but for the sake of a common notation the use of 8% as a measure of the slope is continued. The stability diagrams shown consist of straight lines. A conventional rectangular Cartesian plot gives rise to curved lines and so simplicity and ease of interpretation are achieved by using polar diagrams. Fig. 3 (left-hand part) shows a polar stability and control loss diagram which is typical of those obtained for two-wheel-drive tractors. The line marked BS corresponds to a stability loss at the front wheel B, i.e. B, = 0 (Fig. 2). The point B,/C, is the occurrence of a simultaneous stability loss for wheels B and C, i.e. B, = C, = 0. The line B,/C, to P corresponds to the rear wheel C stability failure, i.e. C, = 0. The extension of line B,/C, to P projected into the quadrant a = 90 to 180” corresponds to the condition C, = 0. Any further increase in /I from the values on this extended line does not result in overturning but merely increases the velocity of descent of what is now a freely rolling three-wheel machine. It is assumed that the differential lock is not engaged. The extended line is the control loss boundary for an infinite coefficient of friction. As a consequence of the tractor being a freely rolling body, with constant acceleration, tipping will not take take place until the condition corresponding to a = 90”, 270” occurs, which is indicated by the
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line PP’. Further details and examples of the stability of freely rolling bodies are given by Reichman. Their stability limits are usually of only academic interest as uncontrollable conditions arise before these limits are reached. This can be seen in Fig. 3 where the control loss boundaries, for tractors X and Y, lie well within the stability boundaries. In general, control loss boundaries are not straight lines. The points on the control loss boundary represent conditions for which controlled straight line motion is not possible and a condition can be any one of those given in Appendix A. The predicted control loss boundaries for the two tractors agree reasonably well with the experimentally determined points at a = 180”, the direct descent direction. The values of p (0.725) and R (0.05 x tractor weight) used in the predictions were measured by considering the equilibrium of the tractors when towed over the ground in a locked and unlocked wheel condition. One feature these diagrams reveal is the ability of the tractors to make controlled ascents of slopes they cannot safely descend. 3.3. Experiments using model tractors Usually the minimum safe operating slope is in the direct descent direction, a = 180”. To examine control loss in any other direction would require driving on slopes steeper than the minimum safe one. This was considered too hazardous so recourse was made to models. This procedure has the advantage that limiting conditions of control and stability can be observed in safety. A model was constructed which completely simulated a four- and a two-wheel drive tractor. The model had a pivoted front axle beam and gear boxes to provide the torque equalization of the differential on each axle. The artificial slope on which it was used consisted of a table with an
Fig. 4. Model tractor and trailer having variable geometry
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aluminium surface which could be angled to the horizontal plane. The model tyres were manufactured from a silicone rubber which gave them a friction coefficient of the order of 0.7 on the table. The model was about & full size but no attempt was made to represent any particular tractor to scale although centre of gravity locations relative to ground and rear axle were geometrically similar to those on full size machines. The prime purpose was to see how well the theory of tractor behaviour could predict model behaviour. Experiments on the tilting table were carried out by placing the model at successive heading angles a and at each steadily increasing the slope of the table surface /I until sliding or overturning occurred. The friction coefficient was determined by measuring /? for a = 180” with all wheels locked, since p = tan j? at onset of sliding. The value of R was determined by towing the model with the surface level.
3.4. Details of model The general arrangement of the model can be seen in Fig. 4 in which it is shown representing a four-wheel drive tractor, a special front axle with torque equalizing gear box being used to simulate the front axle drive differential. This differential can be disconnected when a two-wheel drive machine has to be represented. The design is such that wheel base and track are variable and, by the addition of weight, the centre of gravity position can be altered. A model trailer built along similar lines was also used and is shown attached to the tractor. With the four-wheel drive configuration no attempt was made to simulate the torque transmission across the front axle beam pivot that occurs on some full size designs.
Fig. 5. Comparison of experimentally determined control loss conditions and theoretical predictions for the model tractor with same dimensions but (left) high centre of gravity and (right) lower centre of gravity. 0, x , experimental observations; -, theoretical control loss boundaries: ~ - -, theoretical stability boundaries
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3.5. Results from model experiments on two-wheel drive machines The results from some model experiments, together with the corresponding theoretical predictions, are shown in Figs 5,6 and 7. The models are of four different tractor configurations and two different tractor/trailer combinations. The agreement is typical of that obtained between model experiment and theory. The polar diagrams show the theoretical control and stability boundaries for the tractors. A diagram only shows results for 06a<180” as it is the mirror image of that for 180”
Fig. 6. Comparisons of experimentally determined control loss conditions and theoretical predictions for the model tractor having a forward centre of gravity position but (left) with a wide wheel track and (right) a narrow wheel track. 0, X, experimental observations: -, theoretical control loss boundaries; - -, theoretical stability boundaries
weights above the centre of gravity position. Both configurations had rearward centre of gravity positions. It is typical, for these configurations, that the control loss and stability boundaries coincide at a = 0”. On this heading the front wheel loads become zero on theoretical slopes of 58 % and 68 % respectively. Fig. 6 shows experimentally determined control loss boundaries for the two-wheel drive configuration model having a forward position of centre of gravity but with two different track widths. The left-hand part of the diagram is for a configuration with a wide track and the righthand part a narrow track, otherwise dimensions are the same. With the wide track configuration no stability problems occur at a = go”, as evidenced by the slope on which sliding occurs, i.e.
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/l = 68 %, and stability loss occurs, i.e. p = 93 %. The configuration with the narrow track loses stability at a = 90” as evidenced by the coincidence of the control loss and stability boundaries where /3 = 50 %. For both configurations the minimum safe operating slope occurs for a = 180”. The slight difference between the control loss boundaries for a = 0” and 180”is due to differences in the measured value of ,u at the times the experiments were conducted. The safe descent slope, for these models is of the order of 25 %, whereas they have the ability to climb slopes of 42-45 %. Fig. 7 shows control loss boundaries determined experimentally for the model two-wheel drive tractor with trailer. The tractor configuration was the same as for Fig. 5, right-hand part. The trailer had the same weight and dimensions for each experiment but the left-hand part of Fig. 7 is for a forward position of the trailer centre of gravity and the right-hand part for a rearward
Fig. 7. Comparison of experimentally determined control loss conditions and theoretical predictions for the model tractor and trailer. Tractor dimensions as for Fig. 5 (right). Trailer with forward centre of gravity position (left) and rear centre of gravity position (right). ?? , x , experimental observations: -, theoretical control loss boundaries; --, theoretical stability boundaries
position. As can be seen from the figure the effect of the forward centre of gravity of the trailer is to transfer load to the tractor draw bar causing a stability loss at a = O”, occasioned by both tractor front wheels leaving the ground. The rearward movement of the trailer centre of gravity removes the stability loss at a = 0” but it decreases the safe descent slope of the combination, at a = 180”, from /I = 32 % to j? = 24 %, as shown in Fig. 7, right-hand part. 4. Examples of control and stability loss diagrams
Control loss boundaries obtained by observation of the model are in reasonable agreement with those predicted in each of the 6 experiments. In general a predicted stability boundary lies well outside a control boundary, hence the former has no physical significance and its location cannot be confirmed by experiment. However, on slopes on which prediction indicates that overturning
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in pitch or roll is as likely to happen as sliding the instability of the model was observable. Control and stability boundaries calculations for the full size tractors (Fig. 3) have the same forms as those established for the model (Figs 5, 6 and 7) and, in addition, the single observed value available for each tractor agrees with that predicted. The calculating methods used to obtain the boundaries are considered to be validated to an extent sufficient to allow them to be used to study, at least descriptively, the effects of slope on tractor/implement combinations. Four cases are examined, typical of tractor/implement combinations with which incidents have been known to occur. The cases examined are a two-wheel drive tractor with (1) a flat roller, (2) a rear mounted bale handler, (3) a fully mounted fertilizer spreader, and (4) a single axle silage trailer. In each of the cases examined ~1was taken to be 0.7 and R equal to O-05x tractor weight. A comparison is also made between a two- and a four-wheel drive tractor. 4.1. Tractor withjlat roller For this combination the control loss situation is critical. Fig. 8 shows the control loss boundaries of the tractor alone and the tractor plus a flat roller for tractor X (left-hand part) and tractor Y (right-hand part). In addition the effect of ballasting tractor Y is shown. The addition of a flat roller, to a tractor, markedly reduces the safe descent slope of the machine. The tractor X safe descent slope is reduced from p = 38 % to a = 18 %, at a = 180”, by addition of a 3 ton flat roller. Tractor Y with a 4 ton flat roller suffers a reduction from 39% to 17%. The large
Fig. 8. Control loss diagrams for tractors with rollers. Left, tractor X: - -, tractor alone; tractor with 3 ton roller and no ballast. Right, tractor Y: - (I), tractor alone; - - - - -, tractor witi tractor with 4 ton roller 4 ton roller and no ballast; - (2), tractor alone with rear wheel ballast; -, and rear wheel ballast
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reduction in a safe descent slope is due to a roller not increasing tractor rear wheel normal to ground reaction by load transfer to the tractor draw bar, whilst increasing the overall mass to be braked during descent. The lack of load transfer suggests that rear wheel ballast is required to improve the safe descent slope. The effect of adding rear wheel ballast is shown in Fig. 8 (righthand part). On tractor Y alone it improves the safe descent slope from 39 % to 50% and on tractor Y with the 4 ton roller 17 % to 29 %. A roller also reduces the ascent slope which can be climbed but this effect is less of a safety hazard than the descent cases, in which a dangerous increase of speed occurs. The magnitudes of the reduction in an ascent case, a = 0”, are easily read off Fig. 8. At heading angles other than a = 0” or 180” the situation improves and steeper slopes can be negotiated. For safety, however, operation should be restricted to slopes less than the minimum safe operating slope. 4.2.
Tractor with rear mounted bale handler
Tractor Y with this implement behaves in a manner that is characteristic of tractors with heavy fully mounted loads. Fig. 9 (left-hand part) shows the control loss boundary of tractor Y unloaded and unballasted together with (right-hand part) that of the tractor with standard front ballast carrying (1) a 21 bale and (2) a 42 bale, rear mounted bale handler. As can be seen from the diagram the 42 bale handler is far too large for this tractor, it gives poor longitudinal stability, with p = 10% at a = 0”. The control loss diagram for this configuration, for O”
Fig. 9. Tractor Y with fully mounted bale handler. Left, control loss diagram for tractor alone; right, control loss diagram with front end ballast and loaded with 42 bales (- ---), control loss diagram for 21 bale load (- - - - - -) and stability boundary where different from control loss boundary (-)
180
STABILITY
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CONTROL
ON
SLOPING
GROUND
Tractor withfully mountedfertilizer spreader
The stability (left-hand part) and control loss boundaries (right-hand part) for tractor Y with a typical spreader are shown in Fig. 10. The minimum safe operating slope is determined by the control loss boundary. For the spreader empty this is 39 y0 at a = 150”, whereas fully laden it is 38% at a = 30”. One characteristic that is especially dangerous is the reduction in safe direct descent slope, a = 180” between empty and full. The safe descent slope when full is 49% and when empty 41%. This change in safe descent slope is due to the forward movement of the centre of gravity position during emptying. One danger this emphasizes is that during the spreading operation the safe descent slope is reducing all the time and a slope that was safely descended with the spreader fully loaded may be unsafe when it is empty. This case illustrates the dangers involved in using a machine whose characteristics change during operation.
a=
9o”
: 90”
Fig. 10. Tractor Y with fully mounted fertilizer spreader. L.ef, stability boundaries; right, control loss boundaries; - -, fully laden spreader: -, empty spreader
4.4. Tractor with towed single axle trailer Fig. 11 shows the static stability boundary of tractor Y with a single axle trailer and that for the trailer alone. Also shown is the control loss boundary for the tractor/trailer combination. In this instance the diagram from a = 0” to 360” is shown as an example of a complete diagram. This diagram relates to the case met in practice of a nearly fully loaded silage trailer, e.g. 70 % of volume filled. It shows the minimum safe operating slope to be 27 % at a = 15”. 1The safe descent slope a = 180” is only slightly greater at 29%. The feature of the diagram is the indication of a trailer stability loss when travelling across slope, i.e. on heading a = 90, 270”. This is indicated by the intersection of the trailer stability boundary and the control loss boundary. This failure boundary j3 = 50% will only be of consequence if attempts are made to negotiate slopes steeper than the minimum safe operating slope. The diagram illustrates how operating boundaries are found for tractor-trailed implement combinations.
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Fig. 11. Tractor with 70% full siiage trailer. --“---.-, stability boundary of trailer; -,
-,
stability boundary of tractor with trailer: control loss boundary of tractor with trailer
Fig. 12. Experimental demonstration of dtyerence be#ween two-wheel drive andfour-wheel drive configurations. Left: 0, four-wheel drive tractor alone; IJ, tractor with trailer. Right: x , two-wheel drive tractor alone; + , tractor with trailer
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4.5.
AND
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Comparison of a two- and a four-wheel drive tractor
A comparison of the control boundary for a two-wheel drive tractor with that for the machine differing only in that the front wheels are driven (Fig. 12) shows the descent of slopes to be much safer with the four-wheel drive arrangement. The comparison was made by model experiments. Prediction of control and stability boundaries was not attempted because of the difficulty of proportioning engine torque between front and rear wheels. A tractor has no differential to equalize the distribution. The two- and the four-wheel drive versions of the model were tested with and without the model trailer that was kept identical for each test. The left-hand part of Fig. 12 shows the control loss boundaries of the four-wheel drive tractor with and without the trailer. The right-hand part shows the same for the two-wheel drive machine. The four-wheel drive tractor increases the safe descent slope, a = 180”) from p = 42 % to 68 % for the tractors alone and from /I = 37 % to 50 % for the tractors with trailer. The coincidence of the control loss boundaries for both tractors with trailers for the direct ascent, a = O”, is indicative of the condition being a stability loss as identical two- and four-wheel drive tractors cannot have the same adhesion limited ascent slope at a = 0”. The superior safety characteristics of the fourwheel drive arrangement demonstrated by the experimental results is obtained by utilizing the retarding potential of the front wheels so enabling steeper slopes to be descended.
5.
Conclusions
A tractor will lose stability and start to overturn when the normal to ground component of a wheel load becomes zero. The driver will also lose control when the wheel/ground adhesion is insufficient for equilibrium, in the ground plane, to be maintained. The combinations of slope and direction of travel which result in these occurrences can be calculated by vector mechanics and the boundaries of safe operation represented by polar diagrams. The validity of the method, applied to a two-wheeled tractor, was checked, in the first instance, by reference to downhill slopes on which rear wheel adhesion was insufficient for equilibrium for two different tractors. A more extensive comparison was made by calculating the complete control loss boundaries for 6 different model tractor configurations and comparing these with boundaries obtained by experiment. Operation on hard ground, to which friction concepts apply, was assumed throughout. Reasonable agreement between predicted and observed results was obtained. The methods were used to examine the behaviour on slopes of four widely used tractor and implement combinations. For all the cases examined a control loss limited the slope on which operation was safe, except for the case of the 42 bale handler which so grossly overloaded the tractor that loss of stability occurred on very low slopes. Operation on slopes steeper than the minimum safe operating slope is possible by travel along the contour, e.g. a = go”, 270”. For this direction of travel stability loss may occur before control loss particularly with equipment having a high centre of gravity, e.g. a nearly full silage trailer. Turning from the direction a = 90” or 270” to a downhill direction can be a dangerous manoeuvre. In general the safe descent slope decreases between a = 90”, 270” and a = 180 with the minimum value occurring at a = 180”. This reduction can be very large for some tractor implement combinations of which a tractor with a heavy roller is a common example. The stability boundaries computed take no account of rough ground so it is to be expected that the actual boundaries will occur at smaller values of /I than are indicated in the diagrams. Other risks identified with a two-wheel drive tractor are: (a) the ability to climb slopes it cannot safely descend, and (b) the decrease in the safe descent slope with a fully mounted fertilizer spreader as emptying proceeds. Limited experiments with a model four-wheel drive tractor demonstrated their capability of operating on steeper slopes than those of comparable two-wheel drive machines.
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REFERENCES
Ellis, J. R. Vehicle Dynamics. London: Business Books Ltd., 1969 Reichmann, E. Hangstabilitcit. landwirtschaftlicher Fahrzeuge. ForschungsberichtederBundesversuchsund Prufungsanstolt fur Iandwirtschaftliche Maschinen und Gerate, Wieselburg, Austria, November 1972 Spencer, H. B. Tractor and implement stability on slopes. Scottish Inst. Agric. Engng, Dept. Note SIN/193, August, 1975 (unpubl.) DaskaIov, A. On the dynamic stability of tractors against overturning. Selskostop. Tekh., 1971 8 5 (N.I.A.E. Translation 330) Gilfillan, G.; Spencer H. B. A hazard associated with the braking of two-wheel drive tractors on sloping land. Scottish Inst. Agric. Engng, Dept. Note, SIN/lSO, January 1975 (unpubl.) Smith, D. W.; Liljedahl, J. B. Simulation of the rearward overturning behaviour of a farm tractor. Paper presented at the Midwestern Simulation Council, Moline, Ill., 2 February, 1969 Davis, C. D.; Rebkugler, G. E. Agricultural wheel-tractor overturns-Part II. Mathematical model verifications by scale model study. Trans. A.S.A.E., 1974 484492 KaIoyanov, A.; Stoichev, S.; Sredkov, D. On the static stability of wheeled tractors. Selskostop. Tekh. (Farm Machinery), 1974 11(7) 45-51 Holmes, J. E. Braking force/braking slip measurements over a range of conditions between 0% and 100% sbp. Ministry of Transport Road Research Laboratory Report LR 292, August 1970 Appendix A Methods
of determining
wheel loads
A. 1. Wheel loads on tractor with no external imposed loads The method of determining the wheel loads on a two-wheel drive tractor not acted upon by external loads (Fig. 2) is given by Reichmann .2 His expressions for the wheel loads are the basis of the work reported and are tabulated here for reference. The notation for wheel loads and the tractor geometry used in this paper is identical to that used by Reichmann. The co-ordinates are a body fixed rectangular Cartesian system with the three axes denoted by the numbers 1, 2 and 3 (Fig. 2). Forces are denoted by capital letters, i.e. the vector force on wheel A is denoted by Al where the three components of the force are AI, A,, A,. Position vectors are denoted by lower case letters, i.e. the position of the wheel-ground contact point for wheel A is a, (i = 1, 2, 3). A, =
&lei
(a2_d2)
(bj_a,)
- * &fjlei
{hkei (bj-li)(sk-dk)
(aj- II) (cl-4W2
1 ‘3
=
(d2_dz)
- ii
Et,9el
Qjkei
+(d2-d
(d,_cj)
{[‘%ikel
Wr-~,)
bkmak)
(aj- lj) [(dk- ak) + (Ck-ak)llG2 %kef
(~,-~j) ck-(s2-a2)
et V.,-lj) Gk},
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STABILITY
{h,k
bj-$)
Gk-
AND
CONTROL
4 (G-d,)
ON
SLOPING
GROUND
GA
Assuming that there is no rolling resistance and that the differential lock is not engaged: A,=B,=O,Cz=D,=
-$G,.
On the assumption that Al, B,, Cl and D1 are directly proportional wheels, then B
(Al+Bl)B
=
’ C = (C,+Ddc ’ (C*+D,)
to the normal loads on the
(A,+B,)
”
D = (CI+DI)D ’ (C,+ D3) ”
”
To determine the effect of a trailer or implement on the stability and control of the tractor the additional wheel loads due to a generalized loading Pk of the tractor at a point pk needs to be derived. A.2. Wheel loads due to generalized loud Pk Due to the linearity of the equilibrium equations giving the wheel forces, the effect of a general force Pk can be found by simply adding the wheel forces due to Pk alone to those of the tractor without external imposed loading. The system of forces requiring solution is shown in Fig. 2 with G1=O,&=O,;il=O. The equation of force equilibrium is Ai+B1+Ci+D1+Pt
= 0.
Moment equilibrium is expressed by a&,-‘& + si,kbj& +
‘%,k@k
+
QkdJDk
+
EtJkhPk
=
0.
The moment equation of the rear body about the hinge is &llkel
(Cj
-
II> Ck
+
&JkeI
6-4-b)
Dk
+
&iJkel
(Pj
-
1~) pk
=
0.
Solution of the above equations for the wheel loads due to PI, follows that of Reichmann.a After manipulations solutions for the wheel loads are
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A3= @z-c,) 1P3
(C2-P2)+~2P3~-~3~
C, = Dz = -0.5 Pz, A, = B, = 0. C, = D, as a consequence of the action of the differential unit.
(A,+BJ = A, = (Cl+ D3
=
(d,!a,) 1P,(P,--d,)+P, -
(qy -P2Pl},
(A + 4)
=
B
A
(A,+B,) 3’
’
(Al+B,lB (A3+B3)
3’
&) ($,+d,)-PA+& &-a3), 2
D, =
e2 (c:_dl)
ieel
b2-l2)
+
e2 G-5
p3-p2
-
&I
p3
(P3-l3)1
-PI
(p3
-
Ml
-e3[(P1-1,)P,-P,(p,-1,)1 + (A,+ B3+P3)PI (c2- 12)--e2h- 01 +;[e,(c,+d,-21,)+e,(2Z3)] -(C,+DJ c3
=
=
c
’
[e2(c,-~3)-e3(c2-~2)l,
-P,-A3-B,-D,,
(c,+DJc (C,+D,)
D
=
’
3’
(c,+WD 3’
(C,fD,)
To determine the total wheel forces the above forces due to P, are simply added to those for the tractor without external imposed loading. A.3. Drawbar loads due to trailer In order to solve the equations of the previous section it is required to know the components of Pk acting at the location pk. Considering the equilibrium of the trailer (Fig. 2) P,+BT,+CT,+GT,
= 0,
erjrpt,Pk+ ellkbt,BTk+ q,kct,CTk+ e,,kst,GTk = 0. From these two equations we get &ijk(bt, -
Ct,)
(ptj
-
Ctj)
Pk
+
&Jk
(bt, -
Ct,)
(St,-
Ct,)
GT,
=
0.
. ..(l)
Neglecting rolling resistance BT, = CT, = 0. For conventional trailers bt, = ctg, bt, = ct,. Thus expansion of Eqn (1) gives values of P,. Substitution of the values Pk into the equations derived in section A.2 enables one to find expressions for the wheel forces, N,, N = A, B, C, D, when the forces due to Pk are added to those due to gravity. The expressions are lengthy and computer programs have been written to evaluate the forces.
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A.4. Evaluation of stability and control boundaries Stability loss occurs when the normal to ground force betweenawheelandthegroundvanishes. The values of the wheel forces N,, N = A, B, C, D are used to determine the stability loss points defined by N3 = 0. Control loss occurs when (i) steering control is lost, (ii) rear wheel side slip occurs, (iii) the trailer slides sideways, and (iv) uncontrolled downslope acceleration occurs or uphill travel ceases. When steering control is lost the front wheels are no longer capable of producing the side force necessary for directional equilibrium of the tractor. The condition for steering control loss is
Similarly, rear wheel side slip occurs when
where p1 is the effective friction coefficient in direction 1. The coefficient is evaluated using friction circle c0ncepts.l Sideways slide of a trailer occurs when (BT,+CT,)3p(BT,+CT,). Uncontrolled down slope acceleration occurs when wheel ground adhesion is insufficient at one of the rear wheels for maintenance of equilibrium in the co-ordinate direction 2 during an engine braked descent. Account must be taken of the rolling resistance, R, for this control loss situation. This resistance represents losses occurring during tyre rolling and motion of the differential unit during such control loss situations. Fig. 13 shows diagrammatically the situation Backward /7
spinning wheel
Fig. 13. Criteria of failure. Control loss with engine braking
occurring during such a control loss, with down slope acceleration, under engine braking. Due to the action of the differential, the wheel on which the adhesion is insufficient starts to rotate backwards. If D is the rolling wheel and C the wheel on which there is insufficient adhesion then, equating work done, G,dx = Rdx+ 2puadxC,. Thus the criteria for control loss on wheel C is G&R+2p&
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Similarly, if D is the wheel with insufficient adhesion G,>R+~/L,D~.
Similar arguments apply to uphill travel but the machine is unlikely to slide backwards downhill. Control loss is likely to be characterized by forward motion ceasing as a drive wheel spins. A computer program has been developed to determine the combinations of a and /? at which a control or stability loss occurs. It considers the machine at a fixed heading angle a and increases p in small increments until one of the loss criteria is encountered. To find conditions for control loss the value of p used is that for the ground conditions being studied. To evaluate a stability point the program is re-run with ,U = 106, which effectively represents an infinite coefficient of friction. By taking heading angles between 0” and 360” the complete polar diagram can be constructed. For economy most of the diagrams in this paper are only reproduced for heading angles O”
Fig. 14. Schematic diagram of tractor showing disposition of ballast and mounted implements
Appendix B Details of tractor, implements and ballast used in the stability and control loss examples
A schematic diagram of a two-wheel drive tractor indicating the disposition of loads is shown in Fig. Z4. Dimensions and values of loads are given in Table 1. W = weight of body, D = distance of centre of gravity of body in horizontal plane from rear axle centre line, and H = height of centre of gravity of body from ground measured normal to ground plane. Subscripts refer to body, e.g. W, = weight of body 1. Subscript dejinitions
1 = tractor alone without ballast, 2 = mounted implement, unloaded e.g. bale handler unloaded or for trailed implement centre of gravity position aft of hitch point in a horizontal plane, 3 = mounted implement, loaded e.g. bale handler loaded or for trailed implements position of wheels or rollers aft of hitch point, 4 = ballast on front wheels, 5 = ballast mounted on frame at front of tractor, and 6 = rear wheel ballast consisting of weights and water.
m behind rear axle and m above ground
N.I.A.E. Test Report No. RT 445(1965)
ballasted to produce a forward centre of gravity position
27.2 kg (60 lb): from
All dimensions in m, and weights in kg
$Abnormally
tBaIeweiahtsof
572 1144
Wa --
ground
136 543 3000 4000 6245
152.5 305
W,
0.387m above
1.346 0.840
3400
0.773 0.841
0.722 0.739
HI
2245 2425
Dl
tHitch point taken to be 0.775 m behind rear axle and 0464
*Hitchpointtakento be 0.641
Tractor X* Tractor Yt Balehandler (21 bales)* Balehandler (42 bales)* Tractor Y !j Fertilizer spreader (empty) Fertilizer spreader (full) 3 ton roller 4 ton roller Silage trailer
WI
1.226 1.226 2.0 2.0 3.9
0.965 0.94
De
20 2.0 4.7
1.219 1.702
08
TABLE I
0.72 1.177 0.38 0.38 1641
1.613 1.613
HZ
2.133 2.133
195
W4
-
2.083
1.925 2.083 (MO6
D4
-
255
-
2.083
DS
1.052
Ha
2677
W,
0.840
HB
Stability and Control of Two-wheel Drive Tractors and Machinery on Sloping Ground H. B. SPENCER* This paper examines conditions under which the overturning stability and directional control of two-wheeled drive tractors with mounted or towed implements are lost on sloping ground. Theoretical methods are derived for predicting these conditions. Confirmation of the predictions by experiments on full size machines and models was obtained. The theoretical methods are used to examine the suitability of some frequently used tractor/ implement combinations for use on sloping ground. It is concluded that loss of wheel/ground adhesion, rather than the likelihood of overturning, limits the slope on which most combinations can safely operate.
1. Introduction The stability of a tractor against overturning on sloping ground has received much attention. Overturning does occur on such land but it is unlikely that lack of inherent stability is the primary cause of accidents. It is the author’s contention that accidents on sloping ground are likely to originate from loss of control of the tractor/implement combination followed by an increase in speed. On sloping ground rear wheel adhesion is likely to be lost, especially during descent, and, with retention of front wheel steering control, a dynamically unstable situation arises. If the machine slews an overturn due to centrifugal force can occur due to a high rate of turn.’ This contention that overturning stems in the first instance from control loss is based on an examination of the stability and control of machines operating on sloping ground. The examination, described here, was carried out by the construction of polar diagrams. Polar diagrams were used by Reichmann and Daskalov to show combinations of slope angle, p, and heading angle, a (Fig. 1) at which instability sets in. *s3v* The diagrams have hitherto been confined to vehicles not acted on by external loads and to stability loss, which occurs when the normal to ground reaction of one of the machine’s wheels becomes zero. This paper extends the diagrams to cover instability resulting from external loading and also to take account of wheel/ ground adhesion being inadequate to withstand braking, side or traction forces. Inadequacy results in the driver being unable to drive the tractor on an intended path and is referred to as a control loss. The problems examined are restricted to those arising in steady state straight line motion. Control loss is dependent on the ground conditions. In the absence of sufficient data to provide the general traction/side force/braking characteristic of tyres the work described here is based on friction concepts. This approach is not unduly restrictive since serious control loss problems occur with hard ground conditions for which friction concepts apply.5 The mathematical model describing the control loss boundary, i.e. the critical combinations of slope and heading angles, can easily be extended to cover soft ground conditions once wheel behaviour under these conditions can be described. One feature of the polar diagram depicting the stability and control loss boundaries of a machine is that, in the limit of infinite coefficient of friction, the two are identicalThis is readily seen if one considers that under these conditions control will only be lost when the wheel normal to ground load becomes zero, which corresponds to the stability limit. Extensive use is made of vector algebra to determine expressions for the forces acting on the wheels.* For a vehicle, this technique allows simple formulation of the equilibrium equations for Institute of Agricultural Engineering, Bush Estate, Penicuik, Midlothian, Scotland ‘Scottish Received 16 September 1977; accepted in revised form 8 January 1978
169 OOZI-8634/78/0601~169
$02.00/O
Q 1978 The British Society for Research in Agricultural Engineering
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Fig. 1. Definition of heading angle a and slope angle /I
any geometric configuration, such as a conventional two-wheel drive tractor with a pivoted front axle beam. The equations when solved for the wheel forces, from which the prediction of stability and control loss are made, give rise to lengthy expressions which are evaluated by a computer program. Theoretical predictions of control loss are confirmed by limited tests on two full size tractors and by model experiments on a variety of tractor configurations. The agreement is good enough to warrant use of the techniques for examining circumstances under which control of some commonly used tractor/implement combinations is lost. 2. Prediction of stability and control loss The prediction of combinations of slope /I and heading angle a at which a control or stability loss occurs requires the consideration of the force and moment equilibrium of the tractorimplement combination. This enables wheel ground reactions to be evaluated from which predictions can be made of the machines stability and control loss boundaries. The methods used in determining the wheel ground reactions and criteria for the occurrence of a stability or control loss are given in detail in Appendix A. A general configuration of a two-wheel drive tractor with trailed implement is considered in Fig. 2. The prediction techniques make use of a computer to evaluate the reactions at a fixed heading angle. The slope /3 is increased from zero by small steps until a stability or control loss occurs. By varying the heading angle from 0” to 360” complete stability and control loss diagrams can be determined. 3. Experimental results Experimental confirmation of predicted safety limits on machines poses investigators with problems. Tests on full scale machines to determine limits of safe performance on sloping ground
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KEY TO SYMBOLS A B C D BT CT Qr br Cl d,
denotes tractor wheels
denotes trailer wheels i vector position of wheel/ground
contact points of tractor, m
vector position of wheel/ground
contact points of trailer, m
I
:;: Pt
I-
Ptl Ai Br
vector position of drawbar hitch point on tractor, m vector position of drawbar hitch point on trailer, m vector component
Ci D, CT, Pk
of tractor wheel forces acting at wheel/ground
contact points, kg
1
J
vector component of trailer wheel forces acting at wheel/ground contact points, kg vector component of drawbar hitch load, kg component of unit vector describing the hinge axis of the front axle, m component of a position vector defining a point on the front axle hinge axis, m
;:
weight of front axle inclusive of ballast carried by axle or wheels, kg weight of tractor aft of front axle beam pivot, kg component of the vector giving the position of the centre of gravity of the front axle, m
Sl
; P & i,i,
G, CT,
6,
R
k
component of the vector giving the position of the centre of gravity of tractor excluding front axle, m component of the vector giving the position of the centre of gravity of the complete tractor, m heading angle, degrees slope of land (% slope = 100 x tan /?), % coefficient of sliding friction permutation symbol subscripts taking values 1, 2, 3 vector component of gravity force for tractor, kg vector component of gravity force for trailer, kg vector component of gravity force for front axle, kg rolling resistance, kg
have a high probability of severe accidents occurring involving irreparable damage to expensive machinery or endangering life and limb of operators. Previous investigators have used only one full scale machine in their trials.6 Some investigators have restricted testing to small scale models in order to obtain confirmation of their mathematical models.‘* * The approach used here is to use limited full size trials supplemented by a wide range of tests with small scale models. 3.1. Experiments on full scale tractors By their nature stability and control loss boundaries are not easy to determine
experimentally
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Tractor co-ordinate
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system
k7;
Troller co-ordinate
system
Fig. 2. Co-ordinate system for conventional tractor, with hinged front axle, and trailer
for full size machines. Full size manually controlled experiments to determine control loss can only be conducted in safety for the direct descent case. The requirements are that the slope should not be very long and that a long flat run out area be available at the bottom. Trials were made with two different medium horse power tractors, X and Y, to determine the gradient on which control was lost during direct descent of hard grass slopes. One of the tractors was in the unballasted condition (X). The other (Y) was abnormally ballasted so that control was lost on a relatively moderate slope. Dimensions of tractors X and Y used in these trials are given in Appendix B. The tractors were driven down successively steeper slopes until control was lost. Engine braking, not wheel braking, was used and when wheel/ground adhesion was insufficient for equilibrium the tractor slid bodily downhill with the drive wheels rotating. The suddenness of the loss of control was a feature of these failures. The driver reported that no warning signs were available, such as noticeable wheel slip. This was confirmed from films taken during the trials. It would therefore appear that under hard ground conditions the increase in braking wheelslip attributable to increase in slope under constant engine braking is so small as to be not noticeable by the driver until wheel ground adhesion is insufficient for equilibrium and control is lost. This observation, no noticeable slip, is confirmed by results obtained from testing automobile tyres on hard surfaces.* In the idealized case of a pure friction model with a rigid wheel on hard ground there is no slip at the wheel/ground contact point until the friction force necessary for equilibrium exceeds the maximum available. The dangers inherent in this wheel behaviour are obvious.
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-Fig. 3. Theoretical control and stability loss diagrams for tractor X (lef) (, control loss: stability) and tractor Y (right) (- - - - -, control loss; - -, stability). Experimentally determined control, loss points at a = 180”; ?? , tractor X; x , tractor Y
3.2. Polar diagram representation of stability and control loss The computed stability and control loss boundaries for the two tractors, X and Y, are shown in Fig. 3 together with the experimentally determined control loss points for direct descent. This information is portrayed on a polar diagram with a0 as the circumferential and /I% as the radial co-ordinate. The use of /3 expressed as a ‘A (8% = 100 x tar@) in the polar stability diagram follows Reichmann who showed that the stability boundaries consisted of straight lines in this co-ordinate system which obviously eases construction of such diagrams.2 No such pronouncement on the shape of the control loss boundaries is yet possible but for the sake of a common notation the use of 8% as a measure of the slope is continued. The stability diagrams shown consist of straight lines. A conventional rectangular Cartesian plot gives rise to curved lines and so simplicity and ease of interpretation are achieved by using polar diagrams. Fig. 3 (left-hand part) shows a polar stability and control loss diagram which is typical of those obtained for two-wheel-drive tractors. The line marked BS corresponds to a stability loss at the front wheel B, i.e. B, = 0 (Fig. 2). The point B,/C, is the occurrence of a simultaneous stability loss for wheels B and C, i.e. B, = C, = 0. The line B,/C, to P corresponds to the rear wheel C stability failure, i.e. C, = 0. The extension of line B,/C, to P projected into the quadrant a = 90 to 180” corresponds to the condition C, = 0. Any further increase in /I from the values on this extended line does not result in overturning but merely increases the velocity of descent of what is now a freely rolling three-wheel machine. It is assumed that the differential lock is not engaged. The extended line is the control loss boundary for an infinite coefficient of friction. As a consequence of the tractor being a freely rolling body, with constant acceleration, tipping will not take take place until the condition corresponding to a = 90”, 270” occurs, which is indicated by the
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line PP’. Further details and examples of the stability of freely rolling bodies are given by Reichman. Their stability limits are usually of only academic interest as uncontrollable conditions arise before these limits are reached. This can be seen in Fig. 3 where the control loss boundaries, for tractors X and Y, lie well within the stability boundaries. In general, control loss boundaries are not straight lines. The points on the control loss boundary represent conditions for which controlled straight line motion is not possible and a condition can be any one of those given in Appendix A. The predicted control loss boundaries for the two tractors agree reasonably well with the experimentally determined points at a = 180”, the direct descent direction. The values of p (0.725) and R (0.05 x tractor weight) used in the predictions were measured by considering the equilibrium of the tractors when towed over the ground in a locked and unlocked wheel condition. One feature these diagrams reveal is the ability of the tractors to make controlled ascents of slopes they cannot safely descend. 3.3. Experiments using model tractors Usually the minimum safe operating slope is in the direct descent direction, a = 180”. To examine control loss in any other direction would require driving on slopes steeper than the minimum safe one. This was considered too hazardous so recourse was made to models. This procedure has the advantage that limiting conditions of control and stability can be observed in safety. A model was constructed which completely simulated a four- and a two-wheel drive tractor. The model had a pivoted front axle beam and gear boxes to provide the torque equalization of the differential on each axle. The artificial slope on which it was used consisted of a table with an
Fig. 4. Model tractor and trailer having variable geometry
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175
aluminium surface which could be angled to the horizontal plane. The model tyres were manufactured from a silicone rubber which gave them a friction coefficient of the order of 0.7 on the table. The model was about & full size but no attempt was made to represent any particular tractor to scale although centre of gravity locations relative to ground and rear axle were geometrically similar to those on full size machines. The prime purpose was to see how well the theory of tractor behaviour could predict model behaviour. Experiments on the tilting table were carried out by placing the model at successive heading angles a and at each steadily increasing the slope of the table surface /I until sliding or overturning occurred. The friction coefficient was determined by measuring /? for a = 180” with all wheels locked, since p = tan j? at onset of sliding. The value of R was determined by towing the model with the surface level.
3.4. Details of model The general arrangement of the model can be seen in Fig. 4 in which it is shown representing a four-wheel drive tractor, a special front axle with torque equalizing gear box being used to simulate the front axle drive differential. This differential can be disconnected when a two-wheel drive machine has to be represented. The design is such that wheel base and track are variable and, by the addition of weight, the centre of gravity position can be altered. A model trailer built along similar lines was also used and is shown attached to the tractor. With the four-wheel drive configuration no attempt was made to simulate the torque transmission across the front axle beam pivot that occurs on some full size designs.
Fig. 5. Comparison of experimentally determined control loss conditions and theoretical predictions for the model tractor with same dimensions but (left) high centre of gravity and (right) lower centre of gravity. 0, x , experimental observations; -, theoretical control loss boundaries: ~ - -, theoretical stability boundaries
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3.5. Results from model experiments on two-wheel drive machines The results from some model experiments, together with the corresponding theoretical predictions, are shown in Figs 5,6 and 7. The models are of four different tractor configurations and two different tractor/trailer combinations. The agreement is typical of that obtained between model experiment and theory. The polar diagrams show the theoretical control and stability boundaries for the tractors. A diagram only shows results for 06a<180” as it is the mirror image of that for 180”
Fig. 6. Comparisons of experimentally determined control loss conditions and theoretical predictions for the model tractor having a forward centre of gravity position but (left) with a wide wheel track and (right) a narrow wheel track. 0, X, experimental observations: -, theoretical control loss boundaries; - -, theoretical stability boundaries
weights above the centre of gravity position. Both configurations had rearward centre of gravity positions. It is typical, for these configurations, that the control loss and stability boundaries coincide at a = 0”. On this heading the front wheel loads become zero on theoretical slopes of 58 % and 68 % respectively. Fig. 6 shows experimentally determined control loss boundaries for the two-wheel drive configuration model having a forward position of centre of gravity but with two different track widths. The left-hand part of the diagram is for a configuration with a wide track and the righthand part a narrow track, otherwise dimensions are the same. With the wide track configuration no stability problems occur at a = go”, as evidenced by the slope on which sliding occurs, i.e.
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/l = 68 %, and stability loss occurs, i.e. p = 93 %. The configuration with the narrow track loses stability at a = 90” as evidenced by the coincidence of the control loss and stability boundaries where /3 = 50 %. For both configurations the minimum safe operating slope occurs for a = 180”. The slight difference between the control loss boundaries for a = 0” and 180”is due to differences in the measured value of ,u at the times the experiments were conducted. The safe descent slope, for these models is of the order of 25 %, whereas they have the ability to climb slopes of 42-45 %. Fig. 7 shows control loss boundaries determined experimentally for the model two-wheel drive tractor with trailer. The tractor configuration was the same as for Fig. 5, right-hand part. The trailer had the same weight and dimensions for each experiment but the left-hand part of Fig. 7 is for a forward position of the trailer centre of gravity and the right-hand part for a rearward
Fig. 7. Comparison of experimentally determined control loss conditions and theoretical predictions for the model tractor and trailer. Tractor dimensions as for Fig. 5 (right). Trailer with forward centre of gravity position (left) and rear centre of gravity position (right). ?? , x , experimental observations: -, theoretical control loss boundaries; --, theoretical stability boundaries
position. As can be seen from the figure the effect of the forward centre of gravity of the trailer is to transfer load to the tractor draw bar causing a stability loss at a = O”, occasioned by both tractor front wheels leaving the ground. The rearward movement of the trailer centre of gravity removes the stability loss at a = 0” but it decreases the safe descent slope of the combination, at a = 180”, from /I = 32 % to j? = 24 %, as shown in Fig. 7, right-hand part. 4. Examples of control and stability loss diagrams
Control loss boundaries obtained by observation of the model are in reasonable agreement with those predicted in each of the 6 experiments. In general a predicted stability boundary lies well outside a control boundary, hence the former has no physical significance and its location cannot be confirmed by experiment. However, on slopes on which prediction indicates that overturning
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in pitch or roll is as likely to happen as sliding the instability of the model was observable. Control and stability boundaries calculations for the full size tractors (Fig. 3) have the same forms as those established for the model (Figs 5, 6 and 7) and, in addition, the single observed value available for each tractor agrees with that predicted. The calculating methods used to obtain the boundaries are considered to be validated to an extent sufficient to allow them to be used to study, at least descriptively, the effects of slope on tractor/implement combinations. Four cases are examined, typical of tractor/implement combinations with which incidents have been known to occur. The cases examined are a two-wheel drive tractor with (1) a flat roller, (2) a rear mounted bale handler, (3) a fully mounted fertilizer spreader, and (4) a single axle silage trailer. In each of the cases examined ~1was taken to be 0.7 and R equal to O-05x tractor weight. A comparison is also made between a two- and a four-wheel drive tractor. 4.1. Tractor withjlat roller For this combination the control loss situation is critical. Fig. 8 shows the control loss boundaries of the tractor alone and the tractor plus a flat roller for tractor X (left-hand part) and tractor Y (right-hand part). In addition the effect of ballasting tractor Y is shown. The addition of a flat roller, to a tractor, markedly reduces the safe descent slope of the machine. The tractor X safe descent slope is reduced from p = 38 % to a = 18 %, at a = 180”, by addition of a 3 ton flat roller. Tractor Y with a 4 ton flat roller suffers a reduction from 39% to 17%. The large
Fig. 8. Control loss diagrams for tractors with rollers. Left, tractor X: - -, tractor alone; tractor with 3 ton roller and no ballast. Right, tractor Y: - (I), tractor alone; - - - - -, tractor witi tractor with 4 ton roller 4 ton roller and no ballast; - (2), tractor alone with rear wheel ballast; -, and rear wheel ballast
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reduction in a safe descent slope is due to a roller not increasing tractor rear wheel normal to ground reaction by load transfer to the tractor draw bar, whilst increasing the overall mass to be braked during descent. The lack of load transfer suggests that rear wheel ballast is required to improve the safe descent slope. The effect of adding rear wheel ballast is shown in Fig. 8 (righthand part). On tractor Y alone it improves the safe descent slope from 39 % to 50% and on tractor Y with the 4 ton roller 17 % to 29 %. A roller also reduces the ascent slope which can be climbed but this effect is less of a safety hazard than the descent cases, in which a dangerous increase of speed occurs. The magnitudes of the reduction in an ascent case, a = 0”, are easily read off Fig. 8. At heading angles other than a = 0” or 180” the situation improves and steeper slopes can be negotiated. For safety, however, operation should be restricted to slopes less than the minimum safe operating slope. 4.2.
Tractor with rear mounted bale handler
Tractor Y with this implement behaves in a manner that is characteristic of tractors with heavy fully mounted loads. Fig. 9 (left-hand part) shows the control loss boundary of tractor Y unloaded and unballasted together with (right-hand part) that of the tractor with standard front ballast carrying (1) a 21 bale and (2) a 42 bale, rear mounted bale handler. As can be seen from the diagram the 42 bale handler is far too large for this tractor, it gives poor longitudinal stability, with p = 10% at a = 0”. The control loss diagram for this configuration, for O”
Fig. 9. Tractor Y with fully mounted bale handler. Left, control loss diagram for tractor alone; right, control loss diagram with front end ballast and loaded with 42 bales (- ---), control loss diagram for 21 bale load (- - - - - -) and stability boundary where different from control loss boundary (-)
180
STABILITY
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GROUND
Tractor withfully mountedfertilizer spreader
The stability (left-hand part) and control loss boundaries (right-hand part) for tractor Y with a typical spreader are shown in Fig. 10. The minimum safe operating slope is determined by the control loss boundary. For the spreader empty this is 39 y0 at a = 150”, whereas fully laden it is 38% at a = 30”. One characteristic that is especially dangerous is the reduction in safe direct descent slope, a = 180” between empty and full. The safe descent slope when full is 49% and when empty 41%. This change in safe descent slope is due to the forward movement of the centre of gravity position during emptying. One danger this emphasizes is that during the spreading operation the safe descent slope is reducing all the time and a slope that was safely descended with the spreader fully loaded may be unsafe when it is empty. This case illustrates the dangers involved in using a machine whose characteristics change during operation.
a=
9o”
: 90”
Fig. 10. Tractor Y with fully mounted fertilizer spreader. L.ef, stability boundaries; right, control loss boundaries; - -, fully laden spreader: -, empty spreader
4.4. Tractor with towed single axle trailer Fig. 11 shows the static stability boundary of tractor Y with a single axle trailer and that for the trailer alone. Also shown is the control loss boundary for the tractor/trailer combination. In this instance the diagram from a = 0” to 360” is shown as an example of a complete diagram. This diagram relates to the case met in practice of a nearly fully loaded silage trailer, e.g. 70 % of volume filled. It shows the minimum safe operating slope to be 27 % at a = 15”. 1The safe descent slope a = 180” is only slightly greater at 29%. The feature of the diagram is the indication of a trailer stability loss when travelling across slope, i.e. on heading a = 90, 270”. This is indicated by the intersection of the trailer stability boundary and the control loss boundary. This failure boundary j3 = 50% will only be of consequence if attempts are made to negotiate slopes steeper than the minimum safe operating slope. The diagram illustrates how operating boundaries are found for tractor-trailed implement combinations.
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Fig. 11. Tractor with 70% full siiage trailer. --“---.-, stability boundary of trailer; -,
-,
stability boundary of tractor with trailer: control loss boundary of tractor with trailer
Fig. 12. Experimental demonstration of dtyerence be#ween two-wheel drive andfour-wheel drive configurations. Left: 0, four-wheel drive tractor alone; IJ, tractor with trailer. Right: x , two-wheel drive tractor alone; + , tractor with trailer
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Comparison of a two- and a four-wheel drive tractor
A comparison of the control boundary for a two-wheel drive tractor with that for the machine differing only in that the front wheels are driven (Fig. 12) shows the descent of slopes to be much safer with the four-wheel drive arrangement. The comparison was made by model experiments. Prediction of control and stability boundaries was not attempted because of the difficulty of proportioning engine torque between front and rear wheels. A tractor has no differential to equalize the distribution. The two- and the four-wheel drive versions of the model were tested with and without the model trailer that was kept identical for each test. The left-hand part of Fig. 12 shows the control loss boundaries of the four-wheel drive tractor with and without the trailer. The right-hand part shows the same for the two-wheel drive machine. The four-wheel drive tractor increases the safe descent slope, a = 180”) from p = 42 % to 68 % for the tractors alone and from /I = 37 % to 50 % for the tractors with trailer. The coincidence of the control loss boundaries for both tractors with trailers for the direct ascent, a = O”, is indicative of the condition being a stability loss as identical two- and four-wheel drive tractors cannot have the same adhesion limited ascent slope at a = 0”. The superior safety characteristics of the fourwheel drive arrangement demonstrated by the experimental results is obtained by utilizing the retarding potential of the front wheels so enabling steeper slopes to be descended.
5.
Conclusions
A tractor will lose stability and start to overturn when the normal to ground component of a wheel load becomes zero. The driver will also lose control when the wheel/ground adhesion is insufficient for equilibrium, in the ground plane, to be maintained. The combinations of slope and direction of travel which result in these occurrences can be calculated by vector mechanics and the boundaries of safe operation represented by polar diagrams. The validity of the method, applied to a two-wheeled tractor, was checked, in the first instance, by reference to downhill slopes on which rear wheel adhesion was insufficient for equilibrium for two different tractors. A more extensive comparison was made by calculating the complete control loss boundaries for 6 different model tractor configurations and comparing these with boundaries obtained by experiment. Operation on hard ground, to which friction concepts apply, was assumed throughout. Reasonable agreement between predicted and observed results was obtained. The methods were used to examine the behaviour on slopes of four widely used tractor and implement combinations. For all the cases examined a control loss limited the slope on which operation was safe, except for the case of the 42 bale handler which so grossly overloaded the tractor that loss of stability occurred on very low slopes. Operation on slopes steeper than the minimum safe operating slope is possible by travel along the contour, e.g. a = go”, 270”. For this direction of travel stability loss may occur before control loss particularly with equipment having a high centre of gravity, e.g. a nearly full silage trailer. Turning from the direction a = 90” or 270” to a downhill direction can be a dangerous manoeuvre. In general the safe descent slope decreases between a = 90”, 270” and a = 180 with the minimum value occurring at a = 180”. This reduction can be very large for some tractor implement combinations of which a tractor with a heavy roller is a common example. The stability boundaries computed take no account of rough ground so it is to be expected that the actual boundaries will occur at smaller values of /I than are indicated in the diagrams. Other risks identified with a two-wheel drive tractor are: (a) the ability to climb slopes it cannot safely descend, and (b) the decrease in the safe descent slope with a fully mounted fertilizer spreader as emptying proceeds. Limited experiments with a model four-wheel drive tractor demonstrated their capability of operating on steeper slopes than those of comparable two-wheel drive machines.
H‘. B.
183
SPENCER
REFERENCES
Ellis, J. R. Vehicle Dynamics. London: Business Books Ltd., 1969 Reichmann, E. Hangstabilitcit. landwirtschaftlicher Fahrzeuge. ForschungsberichtederBundesversuchsund Prufungsanstolt fur Iandwirtschaftliche Maschinen und Gerate, Wieselburg, Austria, November 1972 Spencer, H. B. Tractor and implement stability on slopes. Scottish Inst. Agric. Engng, Dept. Note SIN/193, August, 1975 (unpubl.) DaskaIov, A. On the dynamic stability of tractors against overturning. Selskostop. Tekh., 1971 8 5 (N.I.A.E. Translation 330) Gilfillan, G.; Spencer H. B. A hazard associated with the braking of two-wheel drive tractors on sloping land. Scottish Inst. Agric. Engng, Dept. Note, SIN/lSO, January 1975 (unpubl.) Smith, D. W.; Liljedahl, J. B. Simulation of the rearward overturning behaviour of a farm tractor. Paper presented at the Midwestern Simulation Council, Moline, Ill., 2 February, 1969 Davis, C. D.; Rebkugler, G. E. Agricultural wheel-tractor overturns-Part II. Mathematical model verifications by scale model study. Trans. A.S.A.E., 1974 484492 KaIoyanov, A.; Stoichev, S.; Sredkov, D. On the static stability of wheeled tractors. Selskostop. Tekh. (Farm Machinery), 1974 11(7) 45-51 Holmes, J. E. Braking force/braking slip measurements over a range of conditions between 0% and 100% sbp. Ministry of Transport Road Research Laboratory Report LR 292, August 1970 Appendix A Methods
of determining
wheel loads
A. 1. Wheel loads on tractor with no external imposed loads The method of determining the wheel loads on a two-wheel drive tractor not acted upon by external loads (Fig. 2) is given by Reichmann .2 His expressions for the wheel loads are the basis of the work reported and are tabulated here for reference. The notation for wheel loads and the tractor geometry used in this paper is identical to that used by Reichmann. The co-ordinates are a body fixed rectangular Cartesian system with the three axes denoted by the numbers 1, 2 and 3 (Fig. 2). Forces are denoted by capital letters, i.e. the vector force on wheel A is denoted by Al where the three components of the force are AI, A,, A,. Position vectors are denoted by lower case letters, i.e. the position of the wheel-ground contact point for wheel A is a, (i = 1, 2, 3). A, =
&lei
(a2_d2)
(bj_a,)
- * &fjlei
{hkei (bj-li)(sk-dk)
(aj- II) (cl-4W2
1 ‘3
=
(d2_dz)
- ii
Et,9el
Qjkei
+(d2-d
(d,_cj)
{[‘%ikel
Wr-~,)
bkmak)
(aj- lj) [(dk- ak) + (Ck-ak)llG2 %kef
(~,-~j) ck-(s2-a2)
et V.,-lj) Gk},
184
STABILITY
{h,k
bj-$)
Gk-
AND
CONTROL
4 (G-d,)
ON
SLOPING
GROUND
GA
Assuming that there is no rolling resistance and that the differential lock is not engaged: A,=B,=O,Cz=D,=
-$G,.
On the assumption that Al, B,, Cl and D1 are directly proportional wheels, then B
(Al+Bl)B
=
’ C = (C,+Ddc ’ (C*+D,)
to the normal loads on the
(A,+B,)
”
D = (CI+DI)D ’ (C,+ D3) ”
”
To determine the effect of a trailer or implement on the stability and control of the tractor the additional wheel loads due to a generalized loading Pk of the tractor at a point pk needs to be derived. A.2. Wheel loads due to generalized loud Pk Due to the linearity of the equilibrium equations giving the wheel forces, the effect of a general force Pk can be found by simply adding the wheel forces due to Pk alone to those of the tractor without external imposed loading. The system of forces requiring solution is shown in Fig. 2 with G1=O,&=O,;il=O. The equation of force equilibrium is Ai+B1+Ci+D1+Pt
= 0.
Moment equilibrium is expressed by a&,-‘& + si,kbj& +
‘%,k@k
+
QkdJDk
+
EtJkhPk
=
0.
The moment equation of the rear body about the hinge is &llkel
(Cj
-
II> Ck
+
&JkeI
6-4-b)
Dk
+
&iJkel
(Pj
-
1~) pk
=
0.
Solution of the above equations for the wheel loads due to PI, follows that of Reichmann.a After manipulations solutions for the wheel loads are
H.
B.
185
SPENCER
A3= @z-c,) 1P3
(C2-P2)+~2P3~-~3~
C, = Dz = -0.5 Pz, A, = B, = 0. C, = D, as a consequence of the action of the differential unit.
(A,+BJ = A, = (Cl+ D3
=
(d,!a,) 1P,(P,--d,)+P, -
(qy -P2Pl},
(A + 4)
=
B
A
(A,+B,) 3’
’
(Al+B,lB (A3+B3)
3’
&) ($,+d,)-PA+& &-a3), 2
D, =
e2 (c:_dl)
ieel
b2-l2)
+
e2 G-5
p3-p2
-
&I
p3
(P3-l3)1
-PI
(p3
-
Ml
-e3[(P1-1,)P,-P,(p,-1,)1 + (A,+ B3+P3)PI (c2- 12)--e2h- 01 +;[e,(c,+d,-21,)+e,(2Z3)] -(C,+DJ c3
=
=
c
’
[e2(c,-~3)-e3(c2-~2)l,
-P,-A3-B,-D,,
(c,+DJc (C,+D,)
D
=
’
3’
(c,+WD 3’
(C,fD,)
To determine the total wheel forces the above forces due to P, are simply added to those for the tractor without external imposed loading. A.3. Drawbar loads due to trailer In order to solve the equations of the previous section it is required to know the components of Pk acting at the location pk. Considering the equilibrium of the trailer (Fig. 2) P,+BT,+CT,+GT,
= 0,
erjrpt,Pk+ ellkbt,BTk+ q,kct,CTk+ e,,kst,GTk = 0. From these two equations we get &ijk(bt, -
Ct,)
(ptj
-
Ctj)
Pk
+
&Jk
(bt, -
Ct,)
(St,-
Ct,)
GT,
=
0.
. ..(l)
Neglecting rolling resistance BT, = CT, = 0. For conventional trailers bt, = ctg, bt, = ct,. Thus expansion of Eqn (1) gives values of P,. Substitution of the values Pk into the equations derived in section A.2 enables one to find expressions for the wheel forces, N,, N = A, B, C, D, when the forces due to Pk are added to those due to gravity. The expressions are lengthy and computer programs have been written to evaluate the forces.
186
STABILITY
AND
CONTROL
ON
SLOPING
GROUND
A.4. Evaluation of stability and control boundaries Stability loss occurs when the normal to ground force betweenawheelandthegroundvanishes. The values of the wheel forces N,, N = A, B, C, D are used to determine the stability loss points defined by N3 = 0. Control loss occurs when (i) steering control is lost, (ii) rear wheel side slip occurs, (iii) the trailer slides sideways, and (iv) uncontrolled downslope acceleration occurs or uphill travel ceases. When steering control is lost the front wheels are no longer capable of producing the side force necessary for directional equilibrium of the tractor. The condition for steering control loss is
Similarly, rear wheel side slip occurs when
where p1 is the effective friction coefficient in direction 1. The coefficient is evaluated using friction circle c0ncepts.l Sideways slide of a trailer occurs when (BT,+CT,)3p(BT,+CT,). Uncontrolled down slope acceleration occurs when wheel ground adhesion is insufficient at one of the rear wheels for maintenance of equilibrium in the co-ordinate direction 2 during an engine braked descent. Account must be taken of the rolling resistance, R, for this control loss situation. This resistance represents losses occurring during tyre rolling and motion of the differential unit during such control loss situations. Fig. 13 shows diagrammatically the situation Backward /7
spinning wheel
Fig. 13. Criteria of failure. Control loss with engine braking
occurring during such a control loss, with down slope acceleration, under engine braking. Due to the action of the differential, the wheel on which the adhesion is insufficient starts to rotate backwards. If D is the rolling wheel and C the wheel on which there is insufficient adhesion then, equating work done, G,dx = Rdx+ 2puadxC,. Thus the criteria for control loss on wheel C is G&R+2p&
H.
B.
187
SPENCER
Similarly, if D is the wheel with insufficient adhesion G,>R+~/L,D~.
Similar arguments apply to uphill travel but the machine is unlikely to slide backwards downhill. Control loss is likely to be characterized by forward motion ceasing as a drive wheel spins. A computer program has been developed to determine the combinations of a and /? at which a control or stability loss occurs. It considers the machine at a fixed heading angle a and increases p in small increments until one of the loss criteria is encountered. To find conditions for control loss the value of p used is that for the ground conditions being studied. To evaluate a stability point the program is re-run with ,U = 106, which effectively represents an infinite coefficient of friction. By taking heading angles between 0” and 360” the complete polar diagram can be constructed. For economy most of the diagrams in this paper are only reproduced for heading angles O”
Fig. 14. Schematic diagram of tractor showing disposition of ballast and mounted implements
Appendix B Details of tractor, implements and ballast used in the stability and control loss examples
A schematic diagram of a two-wheel drive tractor indicating the disposition of loads is shown in Fig. Z4. Dimensions and values of loads are given in Table 1. W = weight of body, D = distance of centre of gravity of body in horizontal plane from rear axle centre line, and H = height of centre of gravity of body from ground measured normal to ground plane. Subscripts refer to body, e.g. W, = weight of body 1. Subscript dejinitions
1 = tractor alone without ballast, 2 = mounted implement, unloaded e.g. bale handler unloaded or for trailed implement centre of gravity position aft of hitch point in a horizontal plane, 3 = mounted implement, loaded e.g. bale handler loaded or for trailed implements position of wheels or rollers aft of hitch point, 4 = ballast on front wheels, 5 = ballast mounted on frame at front of tractor, and 6 = rear wheel ballast consisting of weights and water.
m behind rear axle and m above ground
N.I.A.E. Test Report No. RT 445(1965)
ballasted to produce a forward centre of gravity position
27.2 kg (60 lb): from
All dimensions in m, and weights in kg
$Abnormally
tBaIeweiahtsof
572 1144
Wa --
ground
136 543 3000 4000 6245
152.5 305
W,
0.387m above
1.346 0.840
3400
0.773 0.841
0.722 0.739
HI
2245 2425
Dl
tHitch point taken to be 0.775 m behind rear axle and 0464
*Hitchpointtakento be 0.641
Tractor X* Tractor Yt Balehandler (21 bales)* Balehandler (42 bales)* Tractor Y !j Fertilizer spreader (empty) Fertilizer spreader (full) 3 ton roller 4 ton roller Silage trailer
WI
1.226 1.226 2.0 2.0 3.9
0.965 0.94
De
20 2.0 4.7
1.219 1.702
08
TABLE I
0.72 1.177 0.38 0.38 1641
1.613 1.613
HZ
2.133 2.133
195
W4
-
2.083
1.925 2.083 (MO6
D4
-
255
-
2.083
DS
1.052
Ha
2677
W,
0.840
HB