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PM—Power and Machinery
Biosystems Engineering (2002) 81(1), 57d71 doi:10.1006/bioe.2001.0007, available online at http://www.idealibrary.com on PM*Power and Machinery
Working Space Requirement for an Agricultural Tractor Axle Suspension Per-Anders Hansson Department of Agricultural Engineering, Swedish University of Agricultural Sciences, P.O. Box 7033, S-750 07 Uppsala, Sweden; e-mail: [email protected] (Received 13 November 2000; accepted in revised form 3 October 2001; published online 19 November 2001)
An agricultural tractor equipped with both front and rear axle suspensions o!ers a number of advantages when compared with a traditional tractor "tted only with seat, cab or front axle suspension. A soft suspension with low natural frequencies normally o!ers better vibration damping capacity than a sti!er one, but needs more travel space to avoid over-travel. The restricted space is therefore the basic constraint when designing e!ective suspensions. The main purpose of this work is to study the link between the need for available suspension travel space and the vibration damping capacity for agricultural tractors supplied with various types of full axle suspensions. In order to study the suspension characteristics, the dynamics of the di!erently designed vehicles are described with linear models, and their movements when driving over standardized tracks are simulated in the time domain. The e!ects of varying linear suspension characteristics on the vibration damping potential and need of travel space are studied. The possibilities of reducing extreme travel values by using non-linear damping elements are also studied and discussed as well as the e!ects and needs of di!erent levelling devices to counteract variations in static load. Simulations have also been performed in order to study the vibration damping potential for vehicles with varying total mass of the rear axle and any coupling devices that may be mounted on the axle. One general conclusion of the work is that a combined use of load levellers and non-linear progressive damper characteristics makes it possible to use a soft suspension with good vibration reduction under normal driving conditions. The non-linear dampers then limit the maximum suspension strokes on rough surfaces and at high speeds, thus avoiding over-travel. Another conclusion is that, for a vehicle with both front and rear axle suspensions and with a normal seat position, the e!ects of the front suspension characteristics are rather limited. The use of a rather sti! front suspension may therefore be appropriate in order to avoid the need of a front load leveller. 2002 Silsoe Research Institute
which is very important on agricultural tractors (Suggs & Huang, 1969). High-speed agricultural tractors are now commercially available and place further increased demands on the suspension technique (Renius, 1994). In all probability, an alternative means of suspension, besides seat suspension, will be required to improve the operator ride comfort on the agricultural working vehicle of the future. However, it seems that the traditional tractor concept, with two big and two small wheels, will also be retained in the forseeable future, and that the
1. Introduction The vibrations to which the driver of an agricultural tractor is subjected are known to be injurious to health and deleterious to performance. The normal way to reduce these vibrations is to mount a vibration damping seat suspension. The main problem with seat suspensions is, however, the relative motion between the driver and the cab. It is also very di$cult to achieve e!ective seat suspension in the horizontal and rotational directions, 1537-5110/02/010057#15 $35.00/0
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Notation c damping coe$cient, N s m\ damping constant in the balanced position, N s m\ c front suspension damping coe$cient, N s m\ QD c rear suspension damping coe$cient, N s m\ QP c front tyre damping coe$cient, N s m\ RD c rear tyre damping coe$cient, N s m\ RP F force from load-levelling device, N f front suspension natural frequency, Hz LQD f rear suspension natural frequency, Hz LQP I frame roll inertia, kg m AV I frame pitch inertia, kg m AW I front axle roll inertia, kg m D? I rear axle roll inertia, kg m P? k integration gain ' k rear suspension spring constant, N m\ QP k front suspension spring constant, N m\ QD k rear tyre spring constant, N m\ RP k front tyre spring constant, N m\ PD l distance from tractor centre of gravity to rear axle, m l distance from tractor centre of gravity to front axle, m l distance from tractor centre line to rear wheel centre, m l distance from tractor centre line to front wheel centre, m l distance from tractor centre line to rear suspen sion mounting point, m l distance from tractor centre of gravity to seat centre point, m l distance from tractor centre of gravity to ground, m l distance from tractor centre of gravity to seat centre point, m l element length, m C c
main work to improve the suspension design should be based on this concept. The general strategy to improve the vibration damping potential of a suspension system is to increase the mass of the suspended parts and thereby to maximize the ratio of sprung to unsprung mass. The suspension principle with highest vibration damping potential is therefore to mount vibration damping elements between the wheel axles and the vehicle frame. This will allow the chassis, cab and operator platform to be included as sprung mass. Other desirable features of the axle suspension concept are (1) the suspension will reduce the loads transmitted directly to the chassis structure, (2) the seat is not
l C m A m D? m P? m QD m QP P P R R QD R QP t
element length in the balanced position, m vehicle frame weight, kg front tyres and front axle weight, kg rear tyres and rear-axle weight, kg load at the front axle, kg load at the rear axle, kg progressivity constant 1, m progressivity constant 2 ratio of damping in the balanced position front suspension damping ratio rear suspension damping ratio time, s
Matrix and state variables used in the simulation model A B V x u v C T C 06 C 07 F F F 0 F 4 F *% R 0% R 0 R 4 R *% R 0%
vehicle dynamics matrix vehicle dynamics matrix vehicle dynamics matrix state variable vector force input vector ground level input vector level for vehicle frame centre of gravity, m vehicle frame roll angle, rad vehicle frame pitch angle, rad damping force in the rear left suspension element, N damping force in the rear right suspension element, N front axle roll angle, rad level for centre of front axle, m ground level at the front left wheel, m ground level at the front right wheel, m rear axle roll angle, rad level for centre of rear axle, m ground level at the rear left wheel, m ground level at the rear right wheel, m
required to provide a large relative movement between the seat and the operator controls, and (3) the suspension ought not to impart a sense of insecurity to the operation as might a cab suspension (Claar et al., 1980). Tractors with front axle suspension have been available for a long time but the e!ects of this construction are mainly limited to reduction of pitch motions and vertical motions caused by the pitch motions. A suspension between the tractor body and the rear wheels may be designed with both rear wheels mounted on a single axle. The three-point hitch and the powertake-o! (p.t.o.) can then be placed on the suspended tractor frame or on the rear wheel axle. The suspension
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may be combined with some type of levelling device to counteract variations in static load. When a heavy rearmounted implement is used, these variations may be big if the three-point hitch is mounted on the tractor chassis. If the coupling devices are mounted on the axle, the demands on the self-levelling device become much smaller. A soft suspension with low natural frequencies normally o!ers better vibration damping capacity than a sti!er one, but needs more travel space to avoid over-travel. The restricted space is therefore the basic constraint when designing e!ective suspensions (Sharp & Crolla, 1987). If the requirement of the suspension for travel space is decreased, the production cost and complexity are also decreased. A suspension with soft springs can be combined with non-linear damping elements to decrease the probability of over-travel in a restricted working space. Agricultural tractor cab suspensions with non-linear damping characteristics in order to limit the travel are are described by Lines et al. (1989) and Hansson (1995). A few theoretical studies have dealt with the vibration damping potential for traditional agricultural tractors with full axle suspension (Claar et al., 1980; Horton & Crolla, 1986). The studies have clearly shown the logical results that tractors with front and rear axle suspensions increase the possibilities to reduce the vibration level when compared with vehicles with only seat, cab or front axle suspensions. However, the studies performed have mainly been focused on the full axle suspension as a concept, and little attention has been paid to the very important connection between vibration damping capacity and need for available travel space when driving on di!erent surfaces. Neither has the in#uence of varying unsuspended mass on the vibration damping capacity been studied. The international standard ISO 2631 for evaluation of human exposure to whole-body vibrations was published in 1974. Revised versions were then published in 1985 and 1997 (ISO, 1997). The human body shows di!erent sensitivity to vibrations depending on direction and frequency content. In the vertical direction, the body's most sensitive range is 4}8 Hz, while sensitivity in the horizontal directions is highest between 1 and 2 Hz. The standard ISO 2631 includes frequency weighting "lters for the vibrations in the three linear directions. A method for measurement and analysis of the e!ects of vibration level on the driver of an agricultural tractor is described in the standard ISO 5008 (ISO, 1979). The standard ISO 5008 also describes two test tracks, one rough and one smooth, designed to imitate normal driving conditions in agriculture. The tracks are useful for comparative studies of di!erent vibration damping systems. The main purpose of this work is to study the link between the need for available suspension travel space
and the vibration damping capacity for agricultural tractors supplied with various types of full axle suspensions. The e!ects of linear and non-linear damping elements and of di!erent levelling devices are studied, and also the e!ects of the unsuspended mass on the vibration damping capacity. In order to study the suspension characteristics, the dynamics of the di!erently designed vehicles are described with linear models, and their movements when driving over standard tracks are simulated in the time domain.
2. The vehicle and suspension characteristics The tractor studied has both rear wheels mounted on a single rear axle connected with springs and dampers to the rest of the vehicle (Fig. 1). The vehicle is also equipped with a front axle suspension mounted between the front axle pivot and the tractor chassis. The rear and front axles have identical track widths. The three-point hitch, the p.t.o., and the drawbar can either be mounted on the rear wheel axle (Fig. 1) or on the suspended tractor frame (Fig. 2). If mounted on the axle, the mass of the unsuspended part becomes rather high. The spring constant k and the damping coe$cient QD c of the single front axle passive linear suspension QD element are related by the front natural frequency f and the front ratio of damping R : LQD QD k "4nm f QD QD LQD
(1)
c "4nR f m QD QD LQD QD
(2)
where m is the mass at the front axle in the balanced QD position and the moment equation: m (l #l )!m l "0 QD A
(3)
ml m " A QD l #l
(4)
gives
where m is the vehicle frame weight, and l and l are the A distances from the centre of gravity to the front and rear axles, respectively. The spring constants k and the damping coe$cients QP c of the two rear axle passive suspension elements are QP calculated so that a de"ned rear natural frequency f LQP and rear damping ratio R are reached: QP k "2nm f QP QP LQP
(5)
c "2nR f m QP QP LQP QP
(6)
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Fig. 1. Schematic view of the tractor with both front and rear axle suspension; c.g., centre of gravity; l1 and l2 , distance from tractor c.g. to rear axle and front axle, respectively; l3 and l4 , distance from tractor centre line to rear and front wheel centres, respectively; l5 , distance from tractor centre line to rear suspension mounting point; l6 , distance from tractor c.g. to seat centre point; l7 and l8 , distance from tractor c.g. to the ground and to the seat centre point, respectively
where m is the mass at the rear axle in the balanced QP position: ml m " A QP l #l
(7)
The non-linear dampers used on the vehicle studied in some of the simulations have a damping coe$cient
Fig. 2. Tractor with the coupling devices located on the vehicle frame
c which is a function of the element travel l : C l !l . C c(l )"c 1# C (8) C P c "R 2nf m (9) LQP QP where l is the length of the element in the balanced C position, and the constants P and P decide the damp ing elements' non-linear characteristics (Fig. 3). The constant P can be de"ned as the deviation from the balanced position where c is twice as high as c . The constant P describes the shape of the curve, the value for P of 2 de"ning a quadratic curve, etc., and R describing the damping ratio in the balanced position. The rear axle suspension may be combined with a function which counteracts static suspension travel dependent on variations in the mass of the suspended parts. The traditional way to do this is to include an electrically controlled highly damped valve to control the pressure in a hydraulic unit in order to obtain a force actuator. The actuator can be mounted to work in parallel with the suspension spring. Such a device, working with low bandwidth, is of relatively low cost and with very reasonable power consumption. This device should not be confused with high bandwidth force actuators needed in a fully active suspension. The load-levelling device is de"ned to work as a simple integrating controller and add a force F parallel with the JJ suspension spring:
F "K JJ J
D
R (lC!lC ) dt
(10)
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Fig. 3. The ratio between the damping constant c and the damping constant in the balanced position c0 for diwerent values of the , P1"0)09 m, P2"1; progressivity constants P1 and P2 ; , P1"0)09 m, P2"3; , P1"0)03 m, P2"1; , P1"0)03 m, P2"3
The characteristics of the device are decided by the integration gain K . '
3. Simulation models and methodology The dynamics of the studied vehicles are described with linear equations of motion (Appendix A). The following assumptions are made when deriving the equations (partially from ElMadany et al., 1979): (1) The cab, engine and tractor chassis is considered as one rigid body, as are each axle with its wheels. (2) All displacements are small. (3) The suspension springs, and springing and damping characteristics of the tyres are described by linear functions of displacement and velocity. (4) The vehicle units are symmetrical about the vertical plane. (5) The wheels remain in contact with the road surface at all times. (6) The road input displacement function is applied to a point at the centre of the tyre contact patch. (7) Non-linearities resulting from suspension stops and dry friction in suspension elements are neglected. The movements of the tractor chassis are modelled in the vertical, pitch and roll degrees of freedom. The vertical and roll movements of the rear and front axles are also included in the model. The tractor is modelled without any attached implement or trailer.
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The tyres are modelled with elasticity in the vertical direction only. More advanced tyre models, describing also the longitudinal and transversal elasticity, may have been used, but for a vehicle with both front and rear axle suspension, the dynamics are very much decided by the suspension element characteristics and the in#uence of tyre dynamics in other directions than the vertical may be neglected in order to decrease model complexity (Crolla & MacLaurin, 1985). Another reason for excluding horizontal tyre characteristics is the limited knowledge about these parameters. It should be observed that this assumption is not valid for unsuspended vehicles such as a traditional agricultural tractor with only seat suspension. The Matlab-based software Simulink (1998), set to implement a Runge}Kutta integration algorithm was used for the simulation. The integration time step was locked to 0)005 s (200 Hz). The two well-known ground pro"les de"ned in ISO 5008 were used as the ground input in the simulations. This standard de"nes a 100 m long smooth track and a rougher 35 m long track. In the standard, the prescribed velocity on the smoother track, which simulates a gently undulating unpaved farm road, is 12 km h\. The prescribed velocity on the rough track, simulating a very rough ploughed "eld, is 5 km h\. As the simulations are done in the time domain, the important e!ects of the correlations between the left and right track and the delay between the input to the front and rear axle are included. The reference point for measurement of vibration levels has been taken to be the driver's seat, which is assumed to be rigidly mounted onto the vehicle frame. All vertical vibration level "gures reported in the study are frequency weighted according to the ISO 2631 standard. All distance and tyre parameters used in the simulations are described in Appendix A. In all studies, except those studying the e!ects of varying front axle suspension characteristics, the front axle suspension natural frequency f has been de"ned to be 1)5 Hz and LQD the ratio of damping to be 1)4. The mass of the front axle m has been de"ned to be D? 400 kg in all studies. The total mass of the unsuspended parts (the rear axle including the "nal gear and the two rear wheels) has been varied in the studies. The mass of the vehicle frame has then also been varied so that the total mass of the vehicle has always been 4000 kg. Horton and Crolla (1986) assume that the pitch inertia is equal to twice the mass of the vehicle frame, while the roll inertia is numerically equal to the mass. They also assume that the roll inertia of a suspended rear axle is numerically equal to the mass. Collins (1991) de"nes a pitch inertia of 6473 kg m and a roll inertia of
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Fig. 4. Vertical vibrations on track 1 and 12 km h!1. The rear natural frequency f is constant for each curve while the LQP damping ratio is varied between 0)4 (X) and 2)0 (O). , f "1)0 Hz; , f "0)6 Hz; f "0)4 Hz; r.m.s., LQP LQP LQP root mean square
Fig. 5. Vertical vibrations on track 2 rear natural frequency f is constant for LQP damping ratio is varied between 0)4 , f "1)0 Hz; , f "0)6 Hz; LQP LQP root mean square
2829 kg m for a tractor with the mass 4060 kg. In this work, the pitch inertia of the chassis has been assumed to be 1)5 times the mass of the part, and the roll inertias of the chassis and the suspended axles numerically equal to the masses.
reduction on the smoother track. However, such a suspension seems to be too soft when driving on a rougher track (track 2) since the suspension travel r.m.s. then becomes approximately 0)03 m which corresponds to a maximum travel of at least 0)10 m. A suspension with a natural frequency as low as 0)4 Hz also places high demands on the load-levelling device, since it corresponds to a static suspension travel of 0)01 m for each 95 N change of load at the axle. The vibration levels in the roll and pitch directions are also in#uenced by the characteristics of the rear suspension elements. Roll and pitch vibrations are not included in ISO 2631 and the acceleration levels in these
4. Simulations 4.1. E+ects of linear suspension characteristics The purpose of the "rst simulations was to study the connection between vibration damping potential and suspension travel for rear axle suspensions with varying linear characteristics. The described model of a vehicle with both front and rear axle suspensions was used. The natural frequency and the ratio of damping of the rear suspension were varied. The total mass of the vehicle was 4000 kg and the rear unsuspended mass was 800 kg. No levelling device was used in these simulations. Figure 4 shows the connection between vertical ISO-weighted vibration level in the cab and suspension travel root mean square (r.m.s.) when driving on the smoother standardized track at 12 km h\. The ratio of damping is varied between 0)2 and 2)0 for the natural frequencies 0)4, 0)6 and 1)0 Hz. Figure 5 shows the results when driving on the rougher standardized track at 5 km h\. The results show, as expected, that the rear axle suspension characteristics have a large e!ect on the vibration level in the cab and that a softer suspension o!ers the best vibration reduction. A linear rear axle suspension with a natural frequency as low as 0)4 Hz, combined with soft dampers, o!ers very good vibration
and 5 km h!1. The each curve while the (X) and 2)0 (O). , f "0)4 Hz; r.m.s., LQP
Fig. 6. Longitudinal X vibrations on track 1 at 12 km h!1. The rear damping ratio R is constant for each curve while the natural QP frequency is varied between 0)4 and 1)5. , R "0)4 Hz; QP , R "0)8 Hz; , R "1)2 Hz; , R "1)6 Hz; QP QP QP , R "2)0 Hz QP
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Fig. 7. Lateral Y vibrations on track 1 at 12 km h!1. The rear damping ratio R is constant for each curve while the natural QP , R "0)4 Hz; frequency is varied between 0)4 and 1)5. QP , R "0)8 Hz; , R "1)2 Hz; , R "1)6 Hz; QP QP QP , R "2)0 Hz QP
Fig. 9. Travel and vibration in the Y direction when driving on track 2 at 5 km h!1. The distance between the tractor centre line and the rear suspension elements l5 is constant for each curve while the rear damping ratio is varied between 0)4 (X ) and 2)0 (O): Upper set of curves represent travel at wheel centre and lower set the element travel; , l5"0)2 m; , l5"0)4 m; , l5"0)6 m; r.m.s., root mean square
directions may be di$cult to interpret. The rotational vibrations are therefore recalculated with reference to longitudinal X and transversal > vibrations using the vertical distance 0)50 m between the centre of gravity (c.g.) of the chassis and the seat. Figures 6 and 7 show the vibration levels in the horizontal directions when driving on the smoother track at 12 km h\. The vehicle parameters are the same as in the previous simulations. Figure 8 shows the vibrations in the > direction when driving on the rougher track at 5 km h\. The vibration level in the X and > directions increase when the near suspension's natural frequency is
increased, but the e!ects of varying suspension characteristics are rather small in the X direction. The vibrations in the > direction show lower levels, but are more in#uenced by the spring and damping characteristics. The transversal vibrations are, for example, reduced by more than 50% when damping ratio is decreased from 2)0 to 0)4.
Fig. 8. Lateral Y vibrations on track 2 at 5 km h!1. The rear damping ratio R is constant for each curve while the natural QP frequency is varied between 0)4 and 1)5. , R "0)4 Hz; QP , R "0)8 Hz; , R "1)2 Hz; , R "1)6 Hz; QP QP QP , R "2)0 Hz QP
4.2. E+ects of the distance between the rear suspension elements The roll sti!ness of the studied suspension is mainly decided by the characteristics of the rear suspension elements and by the distance between the vehicle's central line and each suspension element (l in Fig. 1). The purpose of these simulations was to study the e!ects when this distance between the real elements was varied. It was assumed that the elements are symmetrically mounted. Figure 9 shows the connection between vibration levels in the > direction and rear suspension element travel when driving on track 2 at 5 km h\. The distance between the elements is constant for each curve while the rear damping ratio has been varied between 0)4 and 2)0. The rear natural frequency has been constant 0)5 Hz in all simulations. The travel close to the wheels can be large when using a soft suspension. The travel is therefore also calculated using two imagined suspension elements without spring and damping function, mounted with a horizontal separation of 1)60 m (l "0)8) which is equal to the track width of the vehicle. The connection between the travel of
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Fig. 10. Vertical vibrations on track 1 at 12 km h!1. The rear unsuspended mass m is constant for each curve while the rear QP damping ratio is varied between 0)4 (X) and 2)0 (O): , m "400 kg; , m "800 kg; , m "1200 kg; QP QP QP r.m.s., root mean square
Fig. 11. Suspension travel for diwerently adjusted load levellers at a sudden 400 N change in load for diwerent values of this integra, K "0; , K "!1000; , tion gain K . ' ' ' K "!2000; , K "!4000 ' '
these elements and the vibration levels at the seat is also described in Fig. 8. The results in Fig. 9 show that the distance between the real elements has an important e!ect on the suspension characteristics. For a constant damping ratio, the vibration levels in the > direction are more than doubled when the distance from the centreline to the damper l is increased from 0)2 to 0)8 m. Both the suspension element travel and the vibration levels decrease when l is decreased. However, the analysis of the travel above the tyres shows that a small l increases the movements of the chassis in relation to the rear axle, even if the suspension element travel decreases.
20}30% with the same suspension travel r.m.s. if the unsuspended mass is decreased from 1200 to 400 kg.
4.4. E+ects of load-levelling characteristics In order to decrease the need for available suspension travel space, it is very important to use an e!ectively working levelling device to counteract variations in static load. The purpose of these simulations was therefore to study the e!ects of di!erently adjusted levelling devices. Figure 11 shows the suspension travel for di!erent settings of the integration gain K when, at time t of 1)0 s, '
4.3. E+ects of unsprung mass The purpose of these simulations was to study the in#uence of variations in the rear unsuspended mass on the vibration reduction. The mass of the unsuspended parts has been varied between 400 kg, which may correspond to a rear axle with two very light wheels and without any coupling devices, and 1200 kg which may correspond to a complete rear axle with all coupling devices moving with the axle. The natural frequency of the rear suspension was assumed to be 0)5 Hz in all simulations while the damping ratio was varied. Figure 10 shows the connection between suspension travel and weighted vibration levels in the vertical direction when driving on track 1 at 12 km h\. The results show that the unsuspended mass has a signi"cant in#uence on the vibration levels in the vertical direction. The levels can be decreased by
Fig. 12. Vertical vibration on track 1 at 12 km h!1. The integration gain K is constant for each curve while the damping ratio is ' varied between 0)4 (X) and 2)0 (O). , K "0; ' , K "!1000; , K "!2000; , K "!3000; ' ' ' , K "!4000; r.m.s., root mean square '
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Fig. 13. Vertical vibration and suspension travel on track 1 at 12 km h!1. The progressivity constant P2 is constant for each curve while the progressivity constant P1 is varied. Upper set of curves represent vibration level and lower set suspension , P2"1; , P2"2; , P2"3; r.m.s., root travel. mean square
a sudden 400 N change of the static load was simulated. The vehicle was stationary, the rear suspension's natural frequency was 0)5 Hz and the damping ratio 1)0. The total mass of the vehicle was 4000 kg and the mass of the rear axle 800 kg. Figure 12 shows the connection between suspension travel and vibration level in the vertical direction suspensions with di!erently adjusted load levellers when driving on track 1 at 12 km h\. Also in this case, the natural frequency was 0)5 Hz and the damping ratio was varied. Figure 11 shows that by increasing the integrator gain, the change in static load can be counteracted relatively fast by the load leveller. Figure 12 shows, however, that the increase of the gain has a negative in#uence on the vibration reduction. This is explained as being a result of a situation where, with high ampli"cation in the integrator, the load leveller works as an extra &spring', striving to return the vehicle to the balanced position. The results described in Fig. 11 are highly dependent on the natural frequency of the suspension. If sti!er springs are used, the reaction to a sudden change in load is decreased. As an example, for a suspension with a natural frequency of 1)0 Hz, the maximum travel for a change in load equal to this simulated in Fig. 9 is only 0)0065 m (with K "!2000). ' 4.5. E+ects of non-linear damping elements The purpose of these simulations was to study how di!erent non-linear damping elements a!ected the
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suspension's vibration damping capacity and need for travel space. The characteristics of the dampers are decided by the damping constant in the balanced position c and the two constants P and P . Figure 13 shows the e!ects of variations in P and P on suspension travel r.m.s. and vertical vibration level when driving on track 1 at 12 km h\. The natural frequency of the rear axle suspension was 0)5 Hz and the damping ratio in the balanced position R was 0)5. Over-travel results in very high peak vibration levels in the cab and may cause severe injury to the driver. The simulations described in Fig. 14 study the e!ects of di!erent non-linear damping characteristics on the maximum suspension travel when driving on the rough track 2 at 5 km h\. The non-linear dampers have been de"ned with a value for P of 0)02 m. Results are shown for P equal to 1 and 3, and for normal linear dampers with a constant value for the damping ratio R of 0)5. QP The other parameters in the rear suspension are the same as in the simulations described in Fig. 13. The results in Fig. 13 show that non-linear damping elements can be used to reduce suspension travel. With low P , the travel is more restricted, and the vibration level is higher when the damper progressivity is linear (P "1) compared with when the progressivity is quad ratic or cubic (P "2 or 3). This is explained as being a result of the situation that, when the travel is smaller than the de"ned P , the damping force will be higher with linear progressivity than for quadratic or cubic. For travel bigger than P , dampers with quadratic or cubic progressivity give a higher damping force (see also Fig. 3).
Fig. 14. Maximum suspension travel on track 2 at 5 km h\1. The damping characteristics are constant for each curve while the speed is varied: , linear damping; , progressivity constant P2"1; , P2"3
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Fig. 15. Vibration in the X direction and front suspension travel on track 1 at 12 km h!1. The rear suspension characteristics and the front natural frequency f are constant for each curve while LQD the front damping ratio is varied between 0)4 (X) and 2)0 (O). , f "0)5 Hz; , f "1)0 Hz; , f "1)5 Hz; LQD LQD LQD , f "2)0 Hz; r.m.s., root mean square LQD
Fig. 16. Vibration in the X direction and front suspension travel on track 2 at 5 km h!1. The rear suspension characteristics and the front natural frequency f are constant for each curve while LQD the front damping ratio is varied between 0)4 (X) and 2)0 (O). , f "0)5 Hz; , f "1)0 Hz; , f "1)5 Hz; LQD LQD LQD , f "2)0 Hz; r.m.s., root mean square LQD
For suspensions with high values for P and P equal to 2 or 3, the in#uence of the progressivity is very little during normal driving and the suspension travel and vibration level values are almost the same as for suspensions without progressivity. Figure 14 shows that the progressive dampers very e!ectively reduce the extreme travel values and thereby the risk for over-travel. When driving speed is increased on the rough track 2, the travel r.m.s. increases for all suspensions, but the studied non-linear dampers keep the maximum travel at 6}8 cm, which may be a reasonable value in a practical construction.
between 0)4 and 2)0. The rear natural frequency has been constant at 0)5 Hz and the damping ratio 1)0. The other vehicle parameters have been constant and no levelling device has been used. Figure 17 shows the connection between front suspension travel and vertical vibrations at the seat for the simulations described above and when driving at track 1 at 12 km h\. The longitudinal position of the seat
4.6. E+ects of the front axle suspension characteristics The studied vehicle is designed with both front and rear axle suspensions. In the previous simulations, the front axle suspension parameters have not been changed in order to limit the number of varied parameters. The purpose of these simulations is instead to study the e!ects when the front suspension in varied and the rear suspension is constant. The characteristics of the front suspension mainly in#uence the vertical and pitch vibrations. The pitch vibrations are, as done previously, recalculated to longitudinal vibrations at the driver's seat. Figures 15 and 16 show the connection between front suspension travel and longitudinal vibrations on the two tracks. The front natural frequency has been constant for each curve, while the front damping ratio has been varied
Fig. 17. Vibration in the Z direction and front suspension travel on track 1 and 12 km h!1. The rear suspension characteristics and the natural frequency of the front suspension f are LQD constant for each curve while the front damping ratio is varied between 0)4 (X) and 2)0 (O). Results are shown for two diwerent distances l6 between tractor centre of gravity and seat; , f "1)0 Hz, l6"0)0 m; , f "1)0 Hz, l6"0)8 m; LQD LQD , f "2)0 Hz, l6"0)0 m; , f "2)0 Hz, l6"0)8 m; LQD LQD r.m.s., root mean square
A G RICU L TU RA L T R AC TO R AX LE SU S P EN S I ON
relative to the rear axle may in#uence the vertical vibration levels. The analysis has therefore been performed for two di!erent positions of the seat; the &normal' position 0)40 m in front of the rear axle (the distance from the centre of the seat to the frame c.g. l equal to 0)8 m) and for a position right above the c.g. of the frame 1)20 m in front of the rear axle (l "0)0 m). Figure 17 shows that, for a normal seat position (l "0)8 m), the e!ects of a change in front suspension characteristics on the vertical seat vibrations are small, but become bigger if the seat is moved forward. The e!ects of varying front suspension characteristics on the longitudinal vibrations are more marked, even if the levels at normal driving are rather low. For comparable suspension travel values, the e!ects of a change in the natural frequency of the front suspension from 0)5 to 1)5 Hz increase the longitudinal vibration levels by 10}20%.
5. Discussion The simulations have used the two ground tracks standardized in ISO 5008 as input since they are well known and easy to reproduce. The &average' tractor driving surface probably corresponds rather well to the smoother of the two test tracks, while the characteristics of the rougher track may not be so frequent. This fact suggests that suspension design should be directed at "nding a suspension with good performance on the smoother track rather than on the rougher. An absolute demand on the suspension is, however, that it has to prevent over-travel even on the roughest ground. The design of a suspension with good characteristics on all types of surfaces and speeds is always a problem. The results presented show that a combined use of load levellers, to counteract variations in static travel, and progressive non-linear dampers, makes it possible to design a suspension with high damping potential at normal driving. The non-linear dampers then limit the maximum suspension strokes on rough ground, thereby avoiding over-travel. A technique with even greater potential, but also more expensive, is to adjust the damping characteristics dependent on the driving circumstances, i.e. to design an adaptive suspension. Damping constant variations can then be obtained with a rather simple low bandwidth control of an ori"ce in the dampers (Hansson, 1996). The study of the e!ects of varying load-levelling characteristics shows that it is possible to design a leveller which is able to counteract variations in static load, for example variations in the weight of the driver and possible passengers or variations in levels of #uids, such as hydraulic or diesel oil. Such a leveller can be designed
67
with a low ampli"cation gain (for example with K "!1000 as in Fig. 11) with small e!ects on the ' vibration damping characteristics. The simulations also showed that vehicles with the coupling devices mounted on the rear axle, thereby making it relatively heavy, o!er poorer vibration damping potential compared with vehicles with a lighter rear axle and the coupling devices mounted on the suspended chassis. The results in this work are limited to situations when no implement is mounted to the tractor. This is a simpli"cation since all implements, and especially those mounted to the three-point hitch, will a!ect the dynamics of the tractor to some extent. Another simpli"cation is that the damping elements are modelled without dry friction or other non-linear e!ects causing so-called &slip-stick'. It is also di$cult to design a damper with a constant and non-temperature-dependent relation between element velocity and damping force as is assumed. Tyre characteristics are dependent on a lot of factors, for example driving velocity, in#ation pressure and tyre temperature. According to Kising and GoK hlich (1989) damping decreases substantially with increasing speed and decreases slightly with increasing in#ation pressure. The sti!ness for rolling tyres in general was about 25% lower than for non-rolling tyres. Increasing the in#ation pressure increased the sti!ness. A work by Lines (1987) showed similar results but the damping were in some cases instead increased with increased speed. A sensitivity analysis was done in order to study the e!ects of varying tyre damping parameters on some of the results presented in this work. The analysis showed that the studied parameters in general were very insensitive to changes in the front tyre characteristics. The results presented in Fig. 4 were changed (1% when the front tyre damping was doubled. The results were more sensitive to changes in the rear tyre damping. The acceleration levels presented in Fig. 4 were increased with 11}15% when the rear tyre damping was doubled. The trends in the results were, however, not changed at all. Since it is very di$cult to simulate most of the nonlinear tyre and damping e!ects with reasonable and guaranteed accuracy, there is always a need to perform full-scale experiments to "nally tune the parameters. Full-scale experiments are also necessary to study the characteristics on all types of surfaces, since it is di$cult to use more than a few in simulations The simulation technique will, however, be a very useful tool, when studying the general connection between suspension design and need for travel space. For a normal seat position, the results show that if the vehicle is supplied with both front and rear axle suspensions, the front axle suspension mainly in#uences the longitudinal and pitch vibrations. The vibration levels in
68
P.-A. H A N S S ON
these directions are, however rather low. To simplify the construction, it may therefore be reasonable to use a rather sti! front spring, in order to avoid the need of also having a front load leveller. The studied vehicle was designed with a single rear axle, suspended with two combined spring and damping elements. Simulations have also been performed with a vehicle with each rear wheel mounted on its own individually suspended wheel axle. This construction may be more appropriate for other types of terrain vehicles, for example forestry machines. However, in this case the coupling devices have to be mounted on the suspended vehicle body, which may result in load-levelling problems. The vibration damping characteristics for the studied vehicle with individually suspended rear wheels are, in most situations, somewhat better than for a comparable vehicle with one rear axle, but the di!erences are rather small. Some of the simulations have studied the use of slowreacting self-levelling devices mounted on each suspension spring to counteract for variations in static load. With an extension of the control-unit and a simple gyro, this device can be extended also to get a function whereby the cab is kept horizontal. This function is favourable for example, when ploughing. It will also a!ect the levels of the roll vibrations.
6. Conclusions A rear axle suspension with low damping and low natural frequency generally o!ers very good vibration reduction in the X, > and Z directions, but needs a large travel space to avoid over-travel on rough surfaces. The progressive dampers studied e!ectively reduce the extreme travel values and thereby the risk for over-travel at rough surfaces. The studied non-linear dampers keep the maximum travel at 6}8 cm also at very rough conditions, which may be a reasonable value in a practical construction. The vibration levels in the X and > directions increase when the rear suspension's natural frequency is increased, but the e!ects of varying suspension characteristics are rather small in the X direction. The vibrations in the > direction show lower levels, but are more in#uenced by the spring and damping characteristics. The transverse vibrations are in the simulations, for example, reduced by more than 50% when the damping ratio is decreased from 2)0 to 0)4. The results show that the distance between the rear elements has an important e!ect on the suspension characteristics. For a constant damping ratio, the vibration levels in the > direction are more than doubled when the horizontal separation is increased from 0)4 to 1)6 m. Both
the suspension element travel and the vibration levels decrease when the separation is decreased. However, the analysis of the travel above the tyres shows that a small separation increases the movements of the chassis in relation to the rear axle, even if the suspension element travel decreases. The results also show that the rear unsuspended mass has a signi"cant in#uence on the vibration levels in the vertical direction. The levels can be decreased by 20}30% with the same suspension travel r.m.s. if the unsuspended mass is decreased from 1200 to 400 kg. For a normal seat position, the e!ects of a change in front suspension characteristics on the vertical seat vibrations are small, but become bigger if the seat is moved forward. The e!ects of varying front suspension characteristics on the longitudinal vibrations are more marked, even if the levels at normal driving are rather low. For comparable suspension travel values, the e!ects of a change in the natural frequency of the front suspension from 0)5 to 1)5 Hz increase the longitudinal vibration levels by only 10}20%. A rather sti! front suspension may therefore be appropriate in order to avoid the need of a front load leveller.
References Claar P W; Sheth P N; Buchele W F; Marley S J (1980). Agricultural tractor chassis suspension system for improved ride comfort. ASAE Paper No. 80-1565 Collins T S (1991). Loads in tractor linkages when transporting rear-mounted implements: development of modelling and measurement techniques. Journal of Agricultural Engineering Research, 49, 165}188 Crolla D A; MacLaurin E B (1985). Theoretical and practical aspects of the ride dynamics of o!-road vehicles*Part 1. Journal of Terramechanics, 22, 17}25 ElMadany M M; Dokainish M A; Allan A B (1979). Ride dynamics of articulated vehicles*a literature survey. Vehicle System Dynamics, 8, 287}316 Hansson P-A (1995). Optimization of agricultural tractor cab suspension using the evolution method. Computers and Electronics in Agriculture, 12, 35}49 Hansson P-A (1996). Rear axle suspension with controlled damping on agricultural tractors. Computers and Electronics in Agriculture, 15, 123}147 Horton D N; Crolla D A (1984). Designing o!-road vehicles with good ride-behaviour. In: Proceedings of the Eighth International Society of Terrain Vehicle Systems Conference, Cambridge, UK. Horton D N L; Crolla D A (1986). Theoretical analysis of a semi-active suspension "tted to an o!-road vehicle. Vehicle System Dynamics, 15, 351}372 ISO (1997). Evaluation of human exposure to whole-body vibration. ISO 2631-1, International Organization for Standardization ISO (1979). Agricultural wheeled tractors and "eld machinery*measurements of whole-body vibration at the operator. ISO 5008, International Organization for Standardization
69
A G RICU L TU RA L T R AC TO R AX LE SU S P EN S I ON
Kising A; GoK hlich H (1989). Dynamic characteristics of large tyres. Journal of Agricultural Engineering Research, 43, 11}21 Lines J A (1987). Ride vibration of agricultural tractors: transfer functions between the ground and the tractor body. Journal of Agricultural Engineering Research, 37, 81}91 Lines J A; Peachey R O; Collins T S (1992). Predicting the ride vibration of an unsuspended tractor using the dynamic characteristics of rolling tyres. Journal of Terramechanics, 29, 307}315 Lines J A; Whyte R T; Stayner R M (1989). Agricultural vehicle suspensions: Tractor front axle suspension: In: Proceedings of the Third International Symposium on the International Section of the International Social Security Association for Research on Prevention of Occupational Risks, Vienna Renius K T (1994). Trends in tractor design with particular reference to Europe. Journal of Agricultural Engineering Research, 57, 3}22 Sharp R S; Crolla D A (1987). Road vehicle suspension system design*a review. Vehicle System Dynamics, 16, 167}192 Simulink (1998). User's Guide. The Math Works, Inc, South Natick, MA, USA Suggs C W; Huang B K (1969). Tractor cab suspension design and scale model simulation. Transactions of the ASAE, 12, 283}289
Appendix A: Vehicle dynamics model The tractor with front and rear axle suspension is described with the model: x "Ax#Bu#Vv
(A1)
The state variable vector x is de"ned by x"(R R F F R RQ R RQ F FQ F FQ ) *% 0% *% 0% 0 0 4 4 0 0 4 4 C CQ C CQ C CQ ) (A2) 4 4 06 06 07 07 where R and R are the levels of the ground at the *% 0% left and right rear wheels, F and F are the levels *% 0% of the ground at the left and right front wheels, R and R 0 4 are the level for the centre and the roll angle for the rear axle, F and F are the level for the centre and the roll angle for 0 4 the front axle, and C , C and C are the level for the 4 06 07 centre, the roll angle and the pitch angle for the vehicle frame, respectively. A dot above a variable means time derivative. The system matrices A, B and V are de"ned by (and where T shows that the matrix is transposed):
A(1 : 10, 1 : 18)" 0 0 k l !k l RP RP I I P? P? 0
0
k RP m P? 0
k RP m P? 0
0
0
0
0
0
0
0
0
0* 0 1 !2lk !2l k !2l c RP QP RP I I P? P?
0
0
0
0
0
0
0
0
0
1
0
0
!2c RP m P? 0
0
0
0
1
0
0
0
0
0
0
0)
0
0
0
0
0
!2k !2k RP QP m P? 0
l k RD I D? 0
!l k RD I D? 0
0
0
0
0
0
0
0
0
k RD m D? 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
k RD m D? 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 2k l QP Q I AV 0
0
0
0
0
0
!2k l !2c l RD RD I I D? D? 0 0
0
0
0
0
2k QP m A 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2k l QP I AW
0
0
0
(A3)
70
P.-A. H A N S S ON
A(11 : 18, 1 : 18)" 04* 0
0
0
0
0
0
0
0
0
0
0
0
0
2l k QP I P? 0
0
0
0
0
0
0
0
0
0
0
0
0
0
2k l QP m P? 0
0
0
0
0
0
0
0
0
0
0
0
0
0
!k l QD m D? 0
!c l QD m D? 0
0
0
0
1
k l !2l k QD QP m A 0
c l QD m A 0
0
0
0
0
0
1
0
!k l!2k l QD QP I AW
c l QD I AW
0
0
0
0
2k QP m P? 0
0
0
0
0
0
0
0
1
0
0
!k !2k QD RD m D? 0
!c !2c QD RD m D? 0
k QD m D? 0
c QD m D? 1
k QD m A 0
c QD m A 0
!2k !k QP QD m A 0
!c QD m A 0
0
0
0
0
0
0
0
0
!k l QD I AW
!c l QD I AW
k l !2k l QD QP I AW
c l QD I AW
0* (B)2"
(V)2"
l ! I P?
0
1 ! m P?
0*
l I P?
1 0 ! m P? c l RP 1 0 0 0 0 0 I P? !c l RP 0 0 1 0 0 0 I P?
c RP m P? c RP m P?
0 0 1 0 0
0
0
0
0 0 0 1 0
0
0
0
!2k l QP I AV 0 0
0
l I AV
1 m A
0
l ! I AV
0
0
0
0
0
0
0
0
1 m A
0
l I AW
0
l I AW
0* c l RD 0 0 I D? 0 !c l RD 0 I D?
where k and k are the spring constants of the rear and front RP RD wheels, c and c are the damping constants of the rear and RP RD front wheels, I and I are the roll inertias of the rear and front P? D? axles, m and m are the masses of the rear and front axles, P? D?
0
(A4)
(A5)
(A6)
c RD m D? c RD m D?
and I and I are the roll and pitch inertias of the frame, AV AW respectively. The force input matrix u is de"ned by u"(F F ) (A7)
A G RICU L TU RA L T R AC TO R AX LE SU S P EN S I ON
where F and F are the forces from the left and right rear suspension elements. The ground level input matrix v is de"ned by v"(RQ RQ FQ FQ ) *% 0% *% 0%
(A8)
The damping forces in the rear suspension elements are de"ned by F "!c (!RQ #RQ l #CQ !CQ l #CQ l ) QP 4 0 4 06 07
(A9)
F "!c (!RQ !RQ l #CQ #CQ l #CQ l ) QP 4 0 4 06 07 (10) The parameter values used are (if not other values are de"ned in the text): l "1)2 m, l "1)4 m, l "0)8 m, l "0)8 m,
l "0)5 m, l "0)6 m, l "1)1 m, l "0)5 m, k "350 k N m\, RP c "1000 N s m\, RP k "350 k N m\, RD c "1000 N s m\, RD m "2800 kg, A m "400 kg, D? m "800 kg, P? I "400 kg m, D? l "800 kg m, P? l "2800 kg m, AV l "4200 kg m. AW
71
Working Space Requirement for an Agricultural Tractor Axle Suspension Per-Anders Hansson Department of Agricultural Engineering, Swedish University of Agricultural Sciences, P.O. Box 7033, S-750 07 Uppsala, Sweden; e-mail: [email protected] (Received 13 November 2000; accepted in revised form 3 October 2001; published online 19 November 2001)
An agricultural tractor equipped with both front and rear axle suspensions o!ers a number of advantages when compared with a traditional tractor "tted only with seat, cab or front axle suspension. A soft suspension with low natural frequencies normally o!ers better vibration damping capacity than a sti!er one, but needs more travel space to avoid over-travel. The restricted space is therefore the basic constraint when designing e!ective suspensions. The main purpose of this work is to study the link between the need for available suspension travel space and the vibration damping capacity for agricultural tractors supplied with various types of full axle suspensions. In order to study the suspension characteristics, the dynamics of the di!erently designed vehicles are described with linear models, and their movements when driving over standardized tracks are simulated in the time domain. The e!ects of varying linear suspension characteristics on the vibration damping potential and need of travel space are studied. The possibilities of reducing extreme travel values by using non-linear damping elements are also studied and discussed as well as the e!ects and needs of di!erent levelling devices to counteract variations in static load. Simulations have also been performed in order to study the vibration damping potential for vehicles with varying total mass of the rear axle and any coupling devices that may be mounted on the axle. One general conclusion of the work is that a combined use of load levellers and non-linear progressive damper characteristics makes it possible to use a soft suspension with good vibration reduction under normal driving conditions. The non-linear dampers then limit the maximum suspension strokes on rough surfaces and at high speeds, thus avoiding over-travel. Another conclusion is that, for a vehicle with both front and rear axle suspensions and with a normal seat position, the e!ects of the front suspension characteristics are rather limited. The use of a rather sti! front suspension may therefore be appropriate in order to avoid the need of a front load leveller. 2002 Silsoe Research Institute
which is very important on agricultural tractors (Suggs & Huang, 1969). High-speed agricultural tractors are now commercially available and place further increased demands on the suspension technique (Renius, 1994). In all probability, an alternative means of suspension, besides seat suspension, will be required to improve the operator ride comfort on the agricultural working vehicle of the future. However, it seems that the traditional tractor concept, with two big and two small wheels, will also be retained in the forseeable future, and that the
1. Introduction The vibrations to which the driver of an agricultural tractor is subjected are known to be injurious to health and deleterious to performance. The normal way to reduce these vibrations is to mount a vibration damping seat suspension. The main problem with seat suspensions is, however, the relative motion between the driver and the cab. It is also very di$cult to achieve e!ective seat suspension in the horizontal and rotational directions, 1537-5110/02/010057#15 $35.00/0
57
2002 Silsoe Research Institute
58
P.-A. H A N S S ON
Notation c damping coe$cient, N s m\ damping constant in the balanced position, N s m\ c front suspension damping coe$cient, N s m\ QD c rear suspension damping coe$cient, N s m\ QP c front tyre damping coe$cient, N s m\ RD c rear tyre damping coe$cient, N s m\ RP F force from load-levelling device, N f front suspension natural frequency, Hz LQD f rear suspension natural frequency, Hz LQP I frame roll inertia, kg m AV I frame pitch inertia, kg m AW I front axle roll inertia, kg m D? I rear axle roll inertia, kg m P? k integration gain ' k rear suspension spring constant, N m\ QP k front suspension spring constant, N m\ QD k rear tyre spring constant, N m\ RP k front tyre spring constant, N m\ PD l distance from tractor centre of gravity to rear axle, m l distance from tractor centre of gravity to front axle, m l distance from tractor centre line to rear wheel centre, m l distance from tractor centre line to front wheel centre, m l distance from tractor centre line to rear suspen sion mounting point, m l distance from tractor centre of gravity to seat centre point, m l distance from tractor centre of gravity to ground, m l distance from tractor centre of gravity to seat centre point, m l element length, m C c
main work to improve the suspension design should be based on this concept. The general strategy to improve the vibration damping potential of a suspension system is to increase the mass of the suspended parts and thereby to maximize the ratio of sprung to unsprung mass. The suspension principle with highest vibration damping potential is therefore to mount vibration damping elements between the wheel axles and the vehicle frame. This will allow the chassis, cab and operator platform to be included as sprung mass. Other desirable features of the axle suspension concept are (1) the suspension will reduce the loads transmitted directly to the chassis structure, (2) the seat is not
l C m A m D? m P? m QD m QP P P R R QD R QP t
element length in the balanced position, m vehicle frame weight, kg front tyres and front axle weight, kg rear tyres and rear-axle weight, kg load at the front axle, kg load at the rear axle, kg progressivity constant 1, m progressivity constant 2 ratio of damping in the balanced position front suspension damping ratio rear suspension damping ratio time, s
Matrix and state variables used in the simulation model A B V x u v C T C 06 C 07 F F F 0 F 4 F *% R 0% R 0 R 4 R *% R 0%
vehicle dynamics matrix vehicle dynamics matrix vehicle dynamics matrix state variable vector force input vector ground level input vector level for vehicle frame centre of gravity, m vehicle frame roll angle, rad vehicle frame pitch angle, rad damping force in the rear left suspension element, N damping force in the rear right suspension element, N front axle roll angle, rad level for centre of front axle, m ground level at the front left wheel, m ground level at the front right wheel, m rear axle roll angle, rad level for centre of rear axle, m ground level at the rear left wheel, m ground level at the rear right wheel, m
required to provide a large relative movement between the seat and the operator controls, and (3) the suspension ought not to impart a sense of insecurity to the operation as might a cab suspension (Claar et al., 1980). Tractors with front axle suspension have been available for a long time but the e!ects of this construction are mainly limited to reduction of pitch motions and vertical motions caused by the pitch motions. A suspension between the tractor body and the rear wheels may be designed with both rear wheels mounted on a single axle. The three-point hitch and the powertake-o! (p.t.o.) can then be placed on the suspended tractor frame or on the rear wheel axle. The suspension
59
A G RICU L TU RA L T R AC TO R AX LE SU S P EN S I ON
may be combined with some type of levelling device to counteract variations in static load. When a heavy rearmounted implement is used, these variations may be big if the three-point hitch is mounted on the tractor chassis. If the coupling devices are mounted on the axle, the demands on the self-levelling device become much smaller. A soft suspension with low natural frequencies normally o!ers better vibration damping capacity than a sti!er one, but needs more travel space to avoid over-travel. The restricted space is therefore the basic constraint when designing e!ective suspensions (Sharp & Crolla, 1987). If the requirement of the suspension for travel space is decreased, the production cost and complexity are also decreased. A suspension with soft springs can be combined with non-linear damping elements to decrease the probability of over-travel in a restricted working space. Agricultural tractor cab suspensions with non-linear damping characteristics in order to limit the travel are are described by Lines et al. (1989) and Hansson (1995). A few theoretical studies have dealt with the vibration damping potential for traditional agricultural tractors with full axle suspension (Claar et al., 1980; Horton & Crolla, 1986). The studies have clearly shown the logical results that tractors with front and rear axle suspensions increase the possibilities to reduce the vibration level when compared with vehicles with only seat, cab or front axle suspensions. However, the studies performed have mainly been focused on the full axle suspension as a concept, and little attention has been paid to the very important connection between vibration damping capacity and need for available travel space when driving on di!erent surfaces. Neither has the in#uence of varying unsuspended mass on the vibration damping capacity been studied. The international standard ISO 2631 for evaluation of human exposure to whole-body vibrations was published in 1974. Revised versions were then published in 1985 and 1997 (ISO, 1997). The human body shows di!erent sensitivity to vibrations depending on direction and frequency content. In the vertical direction, the body's most sensitive range is 4}8 Hz, while sensitivity in the horizontal directions is highest between 1 and 2 Hz. The standard ISO 2631 includes frequency weighting "lters for the vibrations in the three linear directions. A method for measurement and analysis of the e!ects of vibration level on the driver of an agricultural tractor is described in the standard ISO 5008 (ISO, 1979). The standard ISO 5008 also describes two test tracks, one rough and one smooth, designed to imitate normal driving conditions in agriculture. The tracks are useful for comparative studies of di!erent vibration damping systems. The main purpose of this work is to study the link between the need for available suspension travel space
and the vibration damping capacity for agricultural tractors supplied with various types of full axle suspensions. The e!ects of linear and non-linear damping elements and of di!erent levelling devices are studied, and also the e!ects of the unsuspended mass on the vibration damping capacity. In order to study the suspension characteristics, the dynamics of the di!erently designed vehicles are described with linear models, and their movements when driving over standard tracks are simulated in the time domain.
2. The vehicle and suspension characteristics The tractor studied has both rear wheels mounted on a single rear axle connected with springs and dampers to the rest of the vehicle (Fig. 1). The vehicle is also equipped with a front axle suspension mounted between the front axle pivot and the tractor chassis. The rear and front axles have identical track widths. The three-point hitch, the p.t.o., and the drawbar can either be mounted on the rear wheel axle (Fig. 1) or on the suspended tractor frame (Fig. 2). If mounted on the axle, the mass of the unsuspended part becomes rather high. The spring constant k and the damping coe$cient QD c of the single front axle passive linear suspension QD element are related by the front natural frequency f and the front ratio of damping R : LQD QD k "4nm f QD QD LQD
(1)
c "4nR f m QD QD LQD QD
(2)
where m is the mass at the front axle in the balanced QD position and the moment equation: m (l #l )!m l "0 QD A
(3)
ml m " A QD l #l
(4)
gives
where m is the vehicle frame weight, and l and l are the A distances from the centre of gravity to the front and rear axles, respectively. The spring constants k and the damping coe$cients QP c of the two rear axle passive suspension elements are QP calculated so that a de"ned rear natural frequency f LQP and rear damping ratio R are reached: QP k "2nm f QP QP LQP
(5)
c "2nR f m QP QP LQP QP
(6)
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P.-A. H A N S S ON
Fig. 1. Schematic view of the tractor with both front and rear axle suspension; c.g., centre of gravity; l1 and l2 , distance from tractor c.g. to rear axle and front axle, respectively; l3 and l4 , distance from tractor centre line to rear and front wheel centres, respectively; l5 , distance from tractor centre line to rear suspension mounting point; l6 , distance from tractor c.g. to seat centre point; l7 and l8 , distance from tractor c.g. to the ground and to the seat centre point, respectively
where m is the mass at the rear axle in the balanced QP position: ml m " A QP l #l
(7)
The non-linear dampers used on the vehicle studied in some of the simulations have a damping coe$cient
Fig. 2. Tractor with the coupling devices located on the vehicle frame
c which is a function of the element travel l : C l !l . C c(l )"c 1# C (8) C P c "R 2nf m (9) LQP QP where l is the length of the element in the balanced C position, and the constants P and P decide the damp ing elements' non-linear characteristics (Fig. 3). The constant P can be de"ned as the deviation from the balanced position where c is twice as high as c . The constant P describes the shape of the curve, the value for P of 2 de"ning a quadratic curve, etc., and R describing the damping ratio in the balanced position. The rear axle suspension may be combined with a function which counteracts static suspension travel dependent on variations in the mass of the suspended parts. The traditional way to do this is to include an electrically controlled highly damped valve to control the pressure in a hydraulic unit in order to obtain a force actuator. The actuator can be mounted to work in parallel with the suspension spring. Such a device, working with low bandwidth, is of relatively low cost and with very reasonable power consumption. This device should not be confused with high bandwidth force actuators needed in a fully active suspension. The load-levelling device is de"ned to work as a simple integrating controller and add a force F parallel with the JJ suspension spring:
F "K JJ J
D
R (lC!lC ) dt
(10)
A G RICU L TU RA L T R AC TO R AX LE SU S P EN S I ON
Fig. 3. The ratio between the damping constant c and the damping constant in the balanced position c0 for diwerent values of the , P1"0)09 m, P2"1; progressivity constants P1 and P2 ; , P1"0)09 m, P2"3; , P1"0)03 m, P2"1; , P1"0)03 m, P2"3
The characteristics of the device are decided by the integration gain K . '
3. Simulation models and methodology The dynamics of the studied vehicles are described with linear equations of motion (Appendix A). The following assumptions are made when deriving the equations (partially from ElMadany et al., 1979): (1) The cab, engine and tractor chassis is considered as one rigid body, as are each axle with its wheels. (2) All displacements are small. (3) The suspension springs, and springing and damping characteristics of the tyres are described by linear functions of displacement and velocity. (4) The vehicle units are symmetrical about the vertical plane. (5) The wheels remain in contact with the road surface at all times. (6) The road input displacement function is applied to a point at the centre of the tyre contact patch. (7) Non-linearities resulting from suspension stops and dry friction in suspension elements are neglected. The movements of the tractor chassis are modelled in the vertical, pitch and roll degrees of freedom. The vertical and roll movements of the rear and front axles are also included in the model. The tractor is modelled without any attached implement or trailer.
61
The tyres are modelled with elasticity in the vertical direction only. More advanced tyre models, describing also the longitudinal and transversal elasticity, may have been used, but for a vehicle with both front and rear axle suspension, the dynamics are very much decided by the suspension element characteristics and the in#uence of tyre dynamics in other directions than the vertical may be neglected in order to decrease model complexity (Crolla & MacLaurin, 1985). Another reason for excluding horizontal tyre characteristics is the limited knowledge about these parameters. It should be observed that this assumption is not valid for unsuspended vehicles such as a traditional agricultural tractor with only seat suspension. The Matlab-based software Simulink (1998), set to implement a Runge}Kutta integration algorithm was used for the simulation. The integration time step was locked to 0)005 s (200 Hz). The two well-known ground pro"les de"ned in ISO 5008 were used as the ground input in the simulations. This standard de"nes a 100 m long smooth track and a rougher 35 m long track. In the standard, the prescribed velocity on the smoother track, which simulates a gently undulating unpaved farm road, is 12 km h\. The prescribed velocity on the rough track, simulating a very rough ploughed "eld, is 5 km h\. As the simulations are done in the time domain, the important e!ects of the correlations between the left and right track and the delay between the input to the front and rear axle are included. The reference point for measurement of vibration levels has been taken to be the driver's seat, which is assumed to be rigidly mounted onto the vehicle frame. All vertical vibration level "gures reported in the study are frequency weighted according to the ISO 2631 standard. All distance and tyre parameters used in the simulations are described in Appendix A. In all studies, except those studying the e!ects of varying front axle suspension characteristics, the front axle suspension natural frequency f has been de"ned to be 1)5 Hz and LQD the ratio of damping to be 1)4. The mass of the front axle m has been de"ned to be D? 400 kg in all studies. The total mass of the unsuspended parts (the rear axle including the "nal gear and the two rear wheels) has been varied in the studies. The mass of the vehicle frame has then also been varied so that the total mass of the vehicle has always been 4000 kg. Horton and Crolla (1986) assume that the pitch inertia is equal to twice the mass of the vehicle frame, while the roll inertia is numerically equal to the mass. They also assume that the roll inertia of a suspended rear axle is numerically equal to the mass. Collins (1991) de"nes a pitch inertia of 6473 kg m and a roll inertia of
62
P.-A. H A N S S ON
Fig. 4. Vertical vibrations on track 1 and 12 km h!1. The rear natural frequency f is constant for each curve while the LQP damping ratio is varied between 0)4 (X) and 2)0 (O). , f "1)0 Hz; , f "0)6 Hz; f "0)4 Hz; r.m.s., LQP LQP LQP root mean square
Fig. 5. Vertical vibrations on track 2 rear natural frequency f is constant for LQP damping ratio is varied between 0)4 , f "1)0 Hz; , f "0)6 Hz; LQP LQP root mean square
2829 kg m for a tractor with the mass 4060 kg. In this work, the pitch inertia of the chassis has been assumed to be 1)5 times the mass of the part, and the roll inertias of the chassis and the suspended axles numerically equal to the masses.
reduction on the smoother track. However, such a suspension seems to be too soft when driving on a rougher track (track 2) since the suspension travel r.m.s. then becomes approximately 0)03 m which corresponds to a maximum travel of at least 0)10 m. A suspension with a natural frequency as low as 0)4 Hz also places high demands on the load-levelling device, since it corresponds to a static suspension travel of 0)01 m for each 95 N change of load at the axle. The vibration levels in the roll and pitch directions are also in#uenced by the characteristics of the rear suspension elements. Roll and pitch vibrations are not included in ISO 2631 and the acceleration levels in these
4. Simulations 4.1. E+ects of linear suspension characteristics The purpose of the "rst simulations was to study the connection between vibration damping potential and suspension travel for rear axle suspensions with varying linear characteristics. The described model of a vehicle with both front and rear axle suspensions was used. The natural frequency and the ratio of damping of the rear suspension were varied. The total mass of the vehicle was 4000 kg and the rear unsuspended mass was 800 kg. No levelling device was used in these simulations. Figure 4 shows the connection between vertical ISO-weighted vibration level in the cab and suspension travel root mean square (r.m.s.) when driving on the smoother standardized track at 12 km h\. The ratio of damping is varied between 0)2 and 2)0 for the natural frequencies 0)4, 0)6 and 1)0 Hz. Figure 5 shows the results when driving on the rougher standardized track at 5 km h\. The results show, as expected, that the rear axle suspension characteristics have a large e!ect on the vibration level in the cab and that a softer suspension o!ers the best vibration reduction. A linear rear axle suspension with a natural frequency as low as 0)4 Hz, combined with soft dampers, o!ers very good vibration
and 5 km h!1. The each curve while the (X) and 2)0 (O). , f "0)4 Hz; r.m.s., LQP
Fig. 6. Longitudinal X vibrations on track 1 at 12 km h!1. The rear damping ratio R is constant for each curve while the natural QP frequency is varied between 0)4 and 1)5. , R "0)4 Hz; QP , R "0)8 Hz; , R "1)2 Hz; , R "1)6 Hz; QP QP QP , R "2)0 Hz QP
A G RICU L TU RA L T R AC TO R AX LE SU S P EN S I ON
63
Fig. 7. Lateral Y vibrations on track 1 at 12 km h!1. The rear damping ratio R is constant for each curve while the natural QP , R "0)4 Hz; frequency is varied between 0)4 and 1)5. QP , R "0)8 Hz; , R "1)2 Hz; , R "1)6 Hz; QP QP QP , R "2)0 Hz QP
Fig. 9. Travel and vibration in the Y direction when driving on track 2 at 5 km h!1. The distance between the tractor centre line and the rear suspension elements l5 is constant for each curve while the rear damping ratio is varied between 0)4 (X ) and 2)0 (O): Upper set of curves represent travel at wheel centre and lower set the element travel; , l5"0)2 m; , l5"0)4 m; , l5"0)6 m; r.m.s., root mean square
directions may be di$cult to interpret. The rotational vibrations are therefore recalculated with reference to longitudinal X and transversal > vibrations using the vertical distance 0)50 m between the centre of gravity (c.g.) of the chassis and the seat. Figures 6 and 7 show the vibration levels in the horizontal directions when driving on the smoother track at 12 km h\. The vehicle parameters are the same as in the previous simulations. Figure 8 shows the vibrations in the > direction when driving on the rougher track at 5 km h\. The vibration level in the X and > directions increase when the near suspension's natural frequency is
increased, but the e!ects of varying suspension characteristics are rather small in the X direction. The vibrations in the > direction show lower levels, but are more in#uenced by the spring and damping characteristics. The transversal vibrations are, for example, reduced by more than 50% when damping ratio is decreased from 2)0 to 0)4.
Fig. 8. Lateral Y vibrations on track 2 at 5 km h!1. The rear damping ratio R is constant for each curve while the natural QP frequency is varied between 0)4 and 1)5. , R "0)4 Hz; QP , R "0)8 Hz; , R "1)2 Hz; , R "1)6 Hz; QP QP QP , R "2)0 Hz QP
4.2. E+ects of the distance between the rear suspension elements The roll sti!ness of the studied suspension is mainly decided by the characteristics of the rear suspension elements and by the distance between the vehicle's central line and each suspension element (l in Fig. 1). The purpose of these simulations was to study the e!ects when this distance between the real elements was varied. It was assumed that the elements are symmetrically mounted. Figure 9 shows the connection between vibration levels in the > direction and rear suspension element travel when driving on track 2 at 5 km h\. The distance between the elements is constant for each curve while the rear damping ratio has been varied between 0)4 and 2)0. The rear natural frequency has been constant 0)5 Hz in all simulations. The travel close to the wheels can be large when using a soft suspension. The travel is therefore also calculated using two imagined suspension elements without spring and damping function, mounted with a horizontal separation of 1)60 m (l "0)8) which is equal to the track width of the vehicle. The connection between the travel of
64
P.-A. H A N S S ON
Fig. 10. Vertical vibrations on track 1 at 12 km h!1. The rear unsuspended mass m is constant for each curve while the rear QP damping ratio is varied between 0)4 (X) and 2)0 (O): , m "400 kg; , m "800 kg; , m "1200 kg; QP QP QP r.m.s., root mean square
Fig. 11. Suspension travel for diwerently adjusted load levellers at a sudden 400 N change in load for diwerent values of this integra, K "0; , K "!1000; , tion gain K . ' ' ' K "!2000; , K "!4000 ' '
these elements and the vibration levels at the seat is also described in Fig. 8. The results in Fig. 9 show that the distance between the real elements has an important e!ect on the suspension characteristics. For a constant damping ratio, the vibration levels in the > direction are more than doubled when the distance from the centreline to the damper l is increased from 0)2 to 0)8 m. Both the suspension element travel and the vibration levels decrease when l is decreased. However, the analysis of the travel above the tyres shows that a small l increases the movements of the chassis in relation to the rear axle, even if the suspension element travel decreases.
20}30% with the same suspension travel r.m.s. if the unsuspended mass is decreased from 1200 to 400 kg.
4.4. E+ects of load-levelling characteristics In order to decrease the need for available suspension travel space, it is very important to use an e!ectively working levelling device to counteract variations in static load. The purpose of these simulations was therefore to study the e!ects of di!erently adjusted levelling devices. Figure 11 shows the suspension travel for di!erent settings of the integration gain K when, at time t of 1)0 s, '
4.3. E+ects of unsprung mass The purpose of these simulations was to study the in#uence of variations in the rear unsuspended mass on the vibration reduction. The mass of the unsuspended parts has been varied between 400 kg, which may correspond to a rear axle with two very light wheels and without any coupling devices, and 1200 kg which may correspond to a complete rear axle with all coupling devices moving with the axle. The natural frequency of the rear suspension was assumed to be 0)5 Hz in all simulations while the damping ratio was varied. Figure 10 shows the connection between suspension travel and weighted vibration levels in the vertical direction when driving on track 1 at 12 km h\. The results show that the unsuspended mass has a signi"cant in#uence on the vibration levels in the vertical direction. The levels can be decreased by
Fig. 12. Vertical vibration on track 1 at 12 km h!1. The integration gain K is constant for each curve while the damping ratio is ' varied between 0)4 (X) and 2)0 (O). , K "0; ' , K "!1000; , K "!2000; , K "!3000; ' ' ' , K "!4000; r.m.s., root mean square '
A G RICU L TU RA L T R AC TO R AX LE SU S P EN S I ON
Fig. 13. Vertical vibration and suspension travel on track 1 at 12 km h!1. The progressivity constant P2 is constant for each curve while the progressivity constant P1 is varied. Upper set of curves represent vibration level and lower set suspension , P2"1; , P2"2; , P2"3; r.m.s., root travel. mean square
a sudden 400 N change of the static load was simulated. The vehicle was stationary, the rear suspension's natural frequency was 0)5 Hz and the damping ratio 1)0. The total mass of the vehicle was 4000 kg and the mass of the rear axle 800 kg. Figure 12 shows the connection between suspension travel and vibration level in the vertical direction suspensions with di!erently adjusted load levellers when driving on track 1 at 12 km h\. Also in this case, the natural frequency was 0)5 Hz and the damping ratio was varied. Figure 11 shows that by increasing the integrator gain, the change in static load can be counteracted relatively fast by the load leveller. Figure 12 shows, however, that the increase of the gain has a negative in#uence on the vibration reduction. This is explained as being a result of a situation where, with high ampli"cation in the integrator, the load leveller works as an extra &spring', striving to return the vehicle to the balanced position. The results described in Fig. 11 are highly dependent on the natural frequency of the suspension. If sti!er springs are used, the reaction to a sudden change in load is decreased. As an example, for a suspension with a natural frequency of 1)0 Hz, the maximum travel for a change in load equal to this simulated in Fig. 9 is only 0)0065 m (with K "!2000). ' 4.5. E+ects of non-linear damping elements The purpose of these simulations was to study how di!erent non-linear damping elements a!ected the
65
suspension's vibration damping capacity and need for travel space. The characteristics of the dampers are decided by the damping constant in the balanced position c and the two constants P and P . Figure 13 shows the e!ects of variations in P and P on suspension travel r.m.s. and vertical vibration level when driving on track 1 at 12 km h\. The natural frequency of the rear axle suspension was 0)5 Hz and the damping ratio in the balanced position R was 0)5. Over-travel results in very high peak vibration levels in the cab and may cause severe injury to the driver. The simulations described in Fig. 14 study the e!ects of di!erent non-linear damping characteristics on the maximum suspension travel when driving on the rough track 2 at 5 km h\. The non-linear dampers have been de"ned with a value for P of 0)02 m. Results are shown for P equal to 1 and 3, and for normal linear dampers with a constant value for the damping ratio R of 0)5. QP The other parameters in the rear suspension are the same as in the simulations described in Fig. 13. The results in Fig. 13 show that non-linear damping elements can be used to reduce suspension travel. With low P , the travel is more restricted, and the vibration level is higher when the damper progressivity is linear (P "1) compared with when the progressivity is quad ratic or cubic (P "2 or 3). This is explained as being a result of the situation that, when the travel is smaller than the de"ned P , the damping force will be higher with linear progressivity than for quadratic or cubic. For travel bigger than P , dampers with quadratic or cubic progressivity give a higher damping force (see also Fig. 3).
Fig. 14. Maximum suspension travel on track 2 at 5 km h\1. The damping characteristics are constant for each curve while the speed is varied: , linear damping; , progressivity constant P2"1; , P2"3
66
P.-A. H A N S S ON
Fig. 15. Vibration in the X direction and front suspension travel on track 1 at 12 km h!1. The rear suspension characteristics and the front natural frequency f are constant for each curve while LQD the front damping ratio is varied between 0)4 (X) and 2)0 (O). , f "0)5 Hz; , f "1)0 Hz; , f "1)5 Hz; LQD LQD LQD , f "2)0 Hz; r.m.s., root mean square LQD
Fig. 16. Vibration in the X direction and front suspension travel on track 2 at 5 km h!1. The rear suspension characteristics and the front natural frequency f are constant for each curve while LQD the front damping ratio is varied between 0)4 (X) and 2)0 (O). , f "0)5 Hz; , f "1)0 Hz; , f "1)5 Hz; LQD LQD LQD , f "2)0 Hz; r.m.s., root mean square LQD
For suspensions with high values for P and P equal to 2 or 3, the in#uence of the progressivity is very little during normal driving and the suspension travel and vibration level values are almost the same as for suspensions without progressivity. Figure 14 shows that the progressive dampers very e!ectively reduce the extreme travel values and thereby the risk for over-travel. When driving speed is increased on the rough track 2, the travel r.m.s. increases for all suspensions, but the studied non-linear dampers keep the maximum travel at 6}8 cm, which may be a reasonable value in a practical construction.
between 0)4 and 2)0. The rear natural frequency has been constant at 0)5 Hz and the damping ratio 1)0. The other vehicle parameters have been constant and no levelling device has been used. Figure 17 shows the connection between front suspension travel and vertical vibrations at the seat for the simulations described above and when driving at track 1 at 12 km h\. The longitudinal position of the seat
4.6. E+ects of the front axle suspension characteristics The studied vehicle is designed with both front and rear axle suspensions. In the previous simulations, the front axle suspension parameters have not been changed in order to limit the number of varied parameters. The purpose of these simulations is instead to study the e!ects when the front suspension in varied and the rear suspension is constant. The characteristics of the front suspension mainly in#uence the vertical and pitch vibrations. The pitch vibrations are, as done previously, recalculated to longitudinal vibrations at the driver's seat. Figures 15 and 16 show the connection between front suspension travel and longitudinal vibrations on the two tracks. The front natural frequency has been constant for each curve, while the front damping ratio has been varied
Fig. 17. Vibration in the Z direction and front suspension travel on track 1 and 12 km h!1. The rear suspension characteristics and the natural frequency of the front suspension f are LQD constant for each curve while the front damping ratio is varied between 0)4 (X) and 2)0 (O). Results are shown for two diwerent distances l6 between tractor centre of gravity and seat; , f "1)0 Hz, l6"0)0 m; , f "1)0 Hz, l6"0)8 m; LQD LQD , f "2)0 Hz, l6"0)0 m; , f "2)0 Hz, l6"0)8 m; LQD LQD r.m.s., root mean square
A G RICU L TU RA L T R AC TO R AX LE SU S P EN S I ON
relative to the rear axle may in#uence the vertical vibration levels. The analysis has therefore been performed for two di!erent positions of the seat; the &normal' position 0)40 m in front of the rear axle (the distance from the centre of the seat to the frame c.g. l equal to 0)8 m) and for a position right above the c.g. of the frame 1)20 m in front of the rear axle (l "0)0 m). Figure 17 shows that, for a normal seat position (l "0)8 m), the e!ects of a change in front suspension characteristics on the vertical seat vibrations are small, but become bigger if the seat is moved forward. The e!ects of varying front suspension characteristics on the longitudinal vibrations are more marked, even if the levels at normal driving are rather low. For comparable suspension travel values, the e!ects of a change in the natural frequency of the front suspension from 0)5 to 1)5 Hz increase the longitudinal vibration levels by 10}20%.
5. Discussion The simulations have used the two ground tracks standardized in ISO 5008 as input since they are well known and easy to reproduce. The &average' tractor driving surface probably corresponds rather well to the smoother of the two test tracks, while the characteristics of the rougher track may not be so frequent. This fact suggests that suspension design should be directed at "nding a suspension with good performance on the smoother track rather than on the rougher. An absolute demand on the suspension is, however, that it has to prevent over-travel even on the roughest ground. The design of a suspension with good characteristics on all types of surfaces and speeds is always a problem. The results presented show that a combined use of load levellers, to counteract variations in static travel, and progressive non-linear dampers, makes it possible to design a suspension with high damping potential at normal driving. The non-linear dampers then limit the maximum suspension strokes on rough ground, thereby avoiding over-travel. A technique with even greater potential, but also more expensive, is to adjust the damping characteristics dependent on the driving circumstances, i.e. to design an adaptive suspension. Damping constant variations can then be obtained with a rather simple low bandwidth control of an ori"ce in the dampers (Hansson, 1996). The study of the e!ects of varying load-levelling characteristics shows that it is possible to design a leveller which is able to counteract variations in static load, for example variations in the weight of the driver and possible passengers or variations in levels of #uids, such as hydraulic or diesel oil. Such a leveller can be designed
67
with a low ampli"cation gain (for example with K "!1000 as in Fig. 11) with small e!ects on the ' vibration damping characteristics. The simulations also showed that vehicles with the coupling devices mounted on the rear axle, thereby making it relatively heavy, o!er poorer vibration damping potential compared with vehicles with a lighter rear axle and the coupling devices mounted on the suspended chassis. The results in this work are limited to situations when no implement is mounted to the tractor. This is a simpli"cation since all implements, and especially those mounted to the three-point hitch, will a!ect the dynamics of the tractor to some extent. Another simpli"cation is that the damping elements are modelled without dry friction or other non-linear e!ects causing so-called &slip-stick'. It is also di$cult to design a damper with a constant and non-temperature-dependent relation between element velocity and damping force as is assumed. Tyre characteristics are dependent on a lot of factors, for example driving velocity, in#ation pressure and tyre temperature. According to Kising and GoK hlich (1989) damping decreases substantially with increasing speed and decreases slightly with increasing in#ation pressure. The sti!ness for rolling tyres in general was about 25% lower than for non-rolling tyres. Increasing the in#ation pressure increased the sti!ness. A work by Lines (1987) showed similar results but the damping were in some cases instead increased with increased speed. A sensitivity analysis was done in order to study the e!ects of varying tyre damping parameters on some of the results presented in this work. The analysis showed that the studied parameters in general were very insensitive to changes in the front tyre characteristics. The results presented in Fig. 4 were changed (1% when the front tyre damping was doubled. The results were more sensitive to changes in the rear tyre damping. The acceleration levels presented in Fig. 4 were increased with 11}15% when the rear tyre damping was doubled. The trends in the results were, however, not changed at all. Since it is very di$cult to simulate most of the nonlinear tyre and damping e!ects with reasonable and guaranteed accuracy, there is always a need to perform full-scale experiments to "nally tune the parameters. Full-scale experiments are also necessary to study the characteristics on all types of surfaces, since it is di$cult to use more than a few in simulations The simulation technique will, however, be a very useful tool, when studying the general connection between suspension design and need for travel space. For a normal seat position, the results show that if the vehicle is supplied with both front and rear axle suspensions, the front axle suspension mainly in#uences the longitudinal and pitch vibrations. The vibration levels in
68
P.-A. H A N S S ON
these directions are, however rather low. To simplify the construction, it may therefore be reasonable to use a rather sti! front spring, in order to avoid the need of also having a front load leveller. The studied vehicle was designed with a single rear axle, suspended with two combined spring and damping elements. Simulations have also been performed with a vehicle with each rear wheel mounted on its own individually suspended wheel axle. This construction may be more appropriate for other types of terrain vehicles, for example forestry machines. However, in this case the coupling devices have to be mounted on the suspended vehicle body, which may result in load-levelling problems. The vibration damping characteristics for the studied vehicle with individually suspended rear wheels are, in most situations, somewhat better than for a comparable vehicle with one rear axle, but the di!erences are rather small. Some of the simulations have studied the use of slowreacting self-levelling devices mounted on each suspension spring to counteract for variations in static load. With an extension of the control-unit and a simple gyro, this device can be extended also to get a function whereby the cab is kept horizontal. This function is favourable for example, when ploughing. It will also a!ect the levels of the roll vibrations.
6. Conclusions A rear axle suspension with low damping and low natural frequency generally o!ers very good vibration reduction in the X, > and Z directions, but needs a large travel space to avoid over-travel on rough surfaces. The progressive dampers studied e!ectively reduce the extreme travel values and thereby the risk for over-travel at rough surfaces. The studied non-linear dampers keep the maximum travel at 6}8 cm also at very rough conditions, which may be a reasonable value in a practical construction. The vibration levels in the X and > directions increase when the rear suspension's natural frequency is increased, but the e!ects of varying suspension characteristics are rather small in the X direction. The vibrations in the > direction show lower levels, but are more in#uenced by the spring and damping characteristics. The transverse vibrations are in the simulations, for example, reduced by more than 50% when the damping ratio is decreased from 2)0 to 0)4. The results show that the distance between the rear elements has an important e!ect on the suspension characteristics. For a constant damping ratio, the vibration levels in the > direction are more than doubled when the horizontal separation is increased from 0)4 to 1)6 m. Both
the suspension element travel and the vibration levels decrease when the separation is decreased. However, the analysis of the travel above the tyres shows that a small separation increases the movements of the chassis in relation to the rear axle, even if the suspension element travel decreases. The results also show that the rear unsuspended mass has a signi"cant in#uence on the vibration levels in the vertical direction. The levels can be decreased by 20}30% with the same suspension travel r.m.s. if the unsuspended mass is decreased from 1200 to 400 kg. For a normal seat position, the e!ects of a change in front suspension characteristics on the vertical seat vibrations are small, but become bigger if the seat is moved forward. The e!ects of varying front suspension characteristics on the longitudinal vibrations are more marked, even if the levels at normal driving are rather low. For comparable suspension travel values, the e!ects of a change in the natural frequency of the front suspension from 0)5 to 1)5 Hz increase the longitudinal vibration levels by only 10}20%. A rather sti! front suspension may therefore be appropriate in order to avoid the need of a front load leveller.
References Claar P W; Sheth P N; Buchele W F; Marley S J (1980). Agricultural tractor chassis suspension system for improved ride comfort. ASAE Paper No. 80-1565 Collins T S (1991). Loads in tractor linkages when transporting rear-mounted implements: development of modelling and measurement techniques. Journal of Agricultural Engineering Research, 49, 165}188 Crolla D A; MacLaurin E B (1985). Theoretical and practical aspects of the ride dynamics of o!-road vehicles*Part 1. Journal of Terramechanics, 22, 17}25 ElMadany M M; Dokainish M A; Allan A B (1979). Ride dynamics of articulated vehicles*a literature survey. Vehicle System Dynamics, 8, 287}316 Hansson P-A (1995). Optimization of agricultural tractor cab suspension using the evolution method. Computers and Electronics in Agriculture, 12, 35}49 Hansson P-A (1996). Rear axle suspension with controlled damping on agricultural tractors. Computers and Electronics in Agriculture, 15, 123}147 Horton D N; Crolla D A (1984). Designing o!-road vehicles with good ride-behaviour. In: Proceedings of the Eighth International Society of Terrain Vehicle Systems Conference, Cambridge, UK. Horton D N L; Crolla D A (1986). Theoretical analysis of a semi-active suspension "tted to an o!-road vehicle. Vehicle System Dynamics, 15, 351}372 ISO (1997). Evaluation of human exposure to whole-body vibration. ISO 2631-1, International Organization for Standardization ISO (1979). Agricultural wheeled tractors and "eld machinery*measurements of whole-body vibration at the operator. ISO 5008, International Organization for Standardization
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Kising A; GoK hlich H (1989). Dynamic characteristics of large tyres. Journal of Agricultural Engineering Research, 43, 11}21 Lines J A (1987). Ride vibration of agricultural tractors: transfer functions between the ground and the tractor body. Journal of Agricultural Engineering Research, 37, 81}91 Lines J A; Peachey R O; Collins T S (1992). Predicting the ride vibration of an unsuspended tractor using the dynamic characteristics of rolling tyres. Journal of Terramechanics, 29, 307}315 Lines J A; Whyte R T; Stayner R M (1989). Agricultural vehicle suspensions: Tractor front axle suspension: In: Proceedings of the Third International Symposium on the International Section of the International Social Security Association for Research on Prevention of Occupational Risks, Vienna Renius K T (1994). Trends in tractor design with particular reference to Europe. Journal of Agricultural Engineering Research, 57, 3}22 Sharp R S; Crolla D A (1987). Road vehicle suspension system design*a review. Vehicle System Dynamics, 16, 167}192 Simulink (1998). User's Guide. The Math Works, Inc, South Natick, MA, USA Suggs C W; Huang B K (1969). Tractor cab suspension design and scale model simulation. Transactions of the ASAE, 12, 283}289
Appendix A: Vehicle dynamics model The tractor with front and rear axle suspension is described with the model: x "Ax#Bu#Vv
(A1)
The state variable vector x is de"ned by x"(R R F F R RQ R RQ F FQ F FQ ) *% 0% *% 0% 0 0 4 4 0 0 4 4 C CQ C CQ C CQ ) (A2) 4 4 06 06 07 07 where R and R are the levels of the ground at the *% 0% left and right rear wheels, F and F are the levels *% 0% of the ground at the left and right front wheels, R and R 0 4 are the level for the centre and the roll angle for the rear axle, F and F are the level for the centre and the roll angle for 0 4 the front axle, and C , C and C are the level for the 4 06 07 centre, the roll angle and the pitch angle for the vehicle frame, respectively. A dot above a variable means time derivative. The system matrices A, B and V are de"ned by (and where T shows that the matrix is transposed):
A(1 : 10, 1 : 18)" 0 0 k l !k l RP RP I I P? P? 0
0
k RP m P? 0
k RP m P? 0
0
0
0
0
0
0
0
0
0* 0 1 !2lk !2l k !2l c RP QP RP I I P? P?
0
0
0
0
0
0
0
0
0
1
0
0
!2c RP m P? 0
0
0
0
1
0
0
0
0
0
0
0)
0
0
0
0
0
!2k !2k RP QP m P? 0
l k RD I D? 0
!l k RD I D? 0
0
0
0
0
0
0
0
0
k RD m D? 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
k RD m D? 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 2k l QP Q I AV 0
0
0
0
0
0
!2k l !2c l RD RD I I D? D? 0 0
0
0
0
0
2k QP m A 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2k l QP I AW
0
0
0
(A3)
70
P.-A. H A N S S ON
A(11 : 18, 1 : 18)" 04* 0
0
0
0
0
0
0
0
0
0
0
0
0
2l k QP I P? 0
0
0
0
0
0
0
0
0
0
0
0
0
0
2k l QP m P? 0
0
0
0
0
0
0
0
0
0
0
0
0
0
!k l QD m D? 0
!c l QD m D? 0
0
0
0
1
k l !2l k QD QP m A 0
c l QD m A 0
0
0
0
0
0
1
0
!k l!2k l QD QP I AW
c l QD I AW
0
0
0
0
2k QP m P? 0
0
0
0
0
0
0
0
1
0
0
!k !2k QD RD m D? 0
!c !2c QD RD m D? 0
k QD m D? 0
c QD m D? 1
k QD m A 0
c QD m A 0
!2k !k QP QD m A 0
!c QD m A 0
0
0
0
0
0
0
0
0
!k l QD I AW
!c l QD I AW
k l !2k l QD QP I AW
c l QD I AW
0* (B)2"
(V)2"
l ! I P?
0
1 ! m P?
0*
l I P?
1 0 ! m P? c l RP 1 0 0 0 0 0 I P? !c l RP 0 0 1 0 0 0 I P?
c RP m P? c RP m P?
0 0 1 0 0
0
0
0
0 0 0 1 0
0
0
0
!2k l QP I AV 0 0
0
l I AV
1 m A
0
l ! I AV
0
0
0
0
0
0
0
0
1 m A
0
l I AW
0
l I AW
0* c l RD 0 0 I D? 0 !c l RD 0 I D?
where k and k are the spring constants of the rear and front RP RD wheels, c and c are the damping constants of the rear and RP RD front wheels, I and I are the roll inertias of the rear and front P? D? axles, m and m are the masses of the rear and front axles, P? D?
0
(A4)
(A5)
(A6)
c RD m D? c RD m D?
and I and I are the roll and pitch inertias of the frame, AV AW respectively. The force input matrix u is de"ned by u"(F F ) (A7)
A G RICU L TU RA L T R AC TO R AX LE SU S P EN S I ON
where F and F are the forces from the left and right rear suspension elements. The ground level input matrix v is de"ned by v"(RQ RQ FQ FQ ) *% 0% *% 0%
(A8)
The damping forces in the rear suspension elements are de"ned by F "!c (!RQ #RQ l #CQ !CQ l #CQ l ) QP 4 0 4 06 07
(A9)
F "!c (!RQ !RQ l #CQ #CQ l #CQ l ) QP 4 0 4 06 07 (10) The parameter values used are (if not other values are de"ned in the text): l "1)2 m, l "1)4 m, l "0)8 m, l "0)8 m,
l "0)5 m, l "0)6 m, l "1)1 m, l "0)5 m, k "350 k N m\, RP c "1000 N s m\, RP k "350 k N m\, RD c "1000 N s m\, RD m "2800 kg, A m "400 kg, D? m "800 kg, P? I "400 kg m, D? l "800 kg m, P? l "2800 kg m, AV l "4200 kg m. AW
71