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Effect of cultivation implements on tractor ride vibration and implications for implement control
J. agric. Engng Res. (1976) 21,247-261
Effect of Cultivation Implements on Tractor Ride Vibration and Implications for Implement Control D. A. CROLLA* The effect of implements on tractor ride vibration is not well understood. Nor is the effect of tractor ride vibration on control of the implement. Previous work on tractor ride dynamics has concentrated on motion of the tractor alone. A four degree of freedom model of a medium power tractor and three furrow plough with soil forces acting on it was developed. The model indicated that the plough exerted considerable damping on the tractor pitch and bounce. As forward speed and tractor vibration levels increased, the damping was likely to become more important. Field results confirmed these predictions. Tractor vertical ride vibration levels, measured with the N.I.A.E. ridemeter were reduced by up to 50% in heavy land and 30% in light land. A five furrow plough reduced tractor ride vibration level in the vertical direction more than a three furrow plough, because the soil forces acting on it were greater. These results have two important implications. Attempts to predict tractor ride vibration should include the effect of the implement or trailer. The damping forces are transmitted to the tractor by the tractor linkage and since conventional draught controls sense linkage force, they receive spurious signals due to the damping forces. These force fluctuations are in the same frequency range as those due to changes in implement depth and cannot therefore be filtered out. The problem increases with speed and if high ploughing speeds i.e. above 2-2 m/s (5 mile/h) are used in the future, an alternative to conventional linkage force sensing controls must be provided.
1. Introduction Previous work on tractor dynamics has considered the tractor alone in attempts to investigate its ride characteristics. This is perhaps surprising since tractors are operated with either an
implement or trailer for most of the time. Some work has been carried out into the behaviour of a tractor and trailer under braking’ but otherwise the effects of implements and trailers on the tractor have been neglected. Recently a survey* of tractor vibration levels for a variety of farm tasks has been carried out and although significant differences were found for various operations, a trend of increasing vibration levels with speed was established. The effect of the implement and draught control response on tractor dynamic performance in the longitudinal direction has been studied3 and a computer model has been devised to simulate the field performance of a tractor operating cultivation implements4. Tractor dynamic motion in the vertical plane however is ignored, tractor vertical displacement being assumed to equal the displacement of the surface contour. This assumption can be questioned at cultivating speeds above 2.2 m/s and it has been suggested that it may result in predicted draught control performance being slightly better than measured performances above 2.2 m/s. If tractor pitch and bounce accelerations are significant their effect will be to increase implement depth variation or if the soil forces prevent depth variation to increase linkage forces and cause a deterioration in draught control performance. Both these effects result in an increased draught variation. In this paper, the equations of motion for a tractor and implement are derived using Lagrange’s equation. The natural frequencies (eigenvalues) and damping ratios are calculated for a medium power tractor with and without a plough. Field measurements, carried out to validate the predicted results are also described. ??N.I.A.E., Wrest Park, Silsoe, Bedford Received 25 July 1975; accepted in t-wised form 23 December 1975
241
248
TRACTOR
RIDE
VIBRATION
LIST OF SYMBOLS a b
distance from rear wheel to tractor c.g. distance from front wheel to tractor
i&ping coefficient of hydraulic lift damping coefficient of sensing unit c2 damping coefficient of front tyre Cf damping coefficient of rear tyre G equivalent damping coefficient of c, soil d implement depth height of tractor c.g. above ground h, hi? height of lower links above ground. H draught force ICCl inertia of implement about its c.g. inertia of implement about centre of 1, gravity of tractor I a2 inertia of implement about lower link ends inertia of tractor about its c.g. J stiffness of hydraulic lift kl stiffness of sensing unit k, stiffness of front tyre k.t stiffness of rear tyre k, Cl
2.
k,
rate of change of draught force with depth II distance from rear wheel to lower link ends I, distance from lower link ends to plough c.g. mast height 1, 14 (a+ll+&) m implement mass M tractor mass ith generalized coordinate 41 4 I external force for particular qr Q coordinate V vertical force on the plough plough velocity VP tractor forward displacement x x, plough forward displacement tractor vertical displacement z plough vertical displacement tractor pitch ! rotation of lower links (relative to 1 tractor) 62 rotation of implement (relative to lower link ends)
Theoretical analysis
2.1. The tractor/implement model Fig. 2 is a diagram of the tractor and implement with the external soil forces acting on it. The four degrees of freedom, or generalized coordinates are, z-tractor bounce, +-tractor pitch, &-rotation of lower links and f3,-rotation of implement. The tractor tyres, compliance in the hydraulic lift and top link sensing unit are all modelled by equivalent springs and dampers. Also the following assumptions are made. (1) The top and lower links are parallel. (2) The lift rods are perpendicular to the lower links. (3) The links have negligible mass compared with that of the tractor or implement. (4) The spring kl is always in tension because the vertical force on the plough is always positive and is therefore considered to be double-acting, as is the damper cl. If the vertical force on the plough was not always positive then the assumption would be invalid because tractor linkages are not restrained in the upward direction. There are two external soil forces acting on the implement, the draught force N and the vertical force V. H is a function of implement depth for a particular field and can be assumed to be proportional to depth over a small range. H=k,(d+z,).
D.
A.
249
CROLLA
V has been found previously3 to be a function of implement depth, rate of change of depth and H Y=f(d+z,)-I-z%. P
The first term,f(d+z,) does not change significantly for small changes in depth, z,, and can be approximated to a constant term V’=f(d). Also, assuming that H remains constant at its value at the mean depth, d, V can be approximated to
and for a constant forward speed v =IV’ - c,i,. The figures used in this analysis refer to a medium power tractor and three furrow mouldboard plough and values of all parameters used are given in Appendix A. DArrows indicate coordinates
genemliscd
b
Fig. 1. Tractor and implement showinggeneralised coordinates and forces (nor to scale)
2.2. Derivation of the equations of motion The method of Lagrange is used to obtain the differential equation of motion of the tractor and implement. The problem must first be expressed in generalized coordinates qt. These are independent of one another and i is the minimum number of them necessary to specify any part of the system. Lagrange’s equation is a differentiation of the energy of the system written in the generalized coordinates . . . (1)
where
= kinetic energy, U = potential energy, K = dissipative energy and Qq t = generalized external force for the q i coordinate. T
This method has advantages for systems with multiple degrees of freedom because only the system energies need to be found. The forces of constraint which do no work but which often complicate the force-mass-acceleration equations are excluded. Also, the calculation of energies
250
TRACTOR
RIDE VIBRATION
requires only the velocities and not the accelerations of the system components which are often more difficult to compute. Substituting each of the generalized coordinates, z, c#, & and 8, into Eqn (1) gives one equation of motion of the system. To calculate the kinetic (T), potential (U) and dissipative (K) energies we need the velocities of the main components. The tractor velocity is simple but the implement velocity, vp, is more complicated because it involves all the generalized coordinates. The vector diagram for the velocity components due to each coordinate is shown in Fig. 2. Hence,
v,2 = i,2 + jcp2 + &cos+-~~ = @,I, COSB,+ ~,i2co~e2 + (&sin e1 + b,l, sin e2 + &, sin$)2
which becomes VP2= &1,2 + 822122 + C$2142 + 2r,1,e,e2 + 21,1,&4 + 21,1,e,Cj+ i2 - i(B,Z, + e2r2+ &) assuming that cos (0, - e,), cos (0, - 4) and cos (0, - $) all equal unity because the angular displacements are small.
Fig. 2.Vector diagram to calculate plough velocity
The energies of the system can now be calculated. Kinetic energy :
T = &Mi2 + iJ& + $mv,” + &
Potential energy :
u = pi&
Dissipative energy :
K = &c,(i + b&” + $cr(.i +
&
+ b#
(&4)2
2
+
+ tJw
-
4;’f &di
q&”
a$@
ic2(~2Q2.
Before substituting these into the Lagrange Eqn (1) the external force Qql for each coordinate must be known.
‘pa)t3In3Im a~ ut?3 (sJopaAua%!a) sadeys apotu pue (sanIsAua@a) sapuanba;r3
aq$ ‘Jauuatu sg
u! uor~oux 30 suoyymba leyIaJa&JgI rapJo-puoDas mo3 aq$ paugqo
Ismltm %U!AITH
.o=~(‘ytl--~q)+z(‘y+~~)+ “yj”3 - ‘9’133 - f cpl”3- ‘3v - Jq) + q (“3+ ‘3 + 4) +
*Molaq ua@ SFQuo uoya.up z ayl.103 aldumxa ut? ‘ss%old @%a~ Jay)tzr B si S%J cmns *wa)sds aql30 uogotu 30 suoymba mo3 a@ 01 awuploo:, pazgelaua% yma 103 (I) ubg II! pa$nl!$sqns aq MOU ku “ba pm x ‘fl ‘J ~03 suoyssaldxa aqJ
‘W9 - ‘1’fI- “19- ZIZlf3+ (“1”e- ‘1%- “I$ - ZX”Y + PZ)“Y- = ““d “se (“1~- Vz + p + %M -3 = =-w voM zee luawaDt?Ids!p B .103puo
tee ~uaumqds!p *Cize- Ye - “if - W3
t? ~03 s!sAp~~a.~e~!tu!sE %upn
+ (zize - IlIe - % - 3 (“Y+ PZYY -
= %
‘srnlaj laplo-puo9as os~e pm alo3aq se suna) ~uersuo:, Bu!~ou81
-ydg3 - ,A) - tdz + p + “?I)(“z + pyy -
= pa /i pm
saA!z
H ~03 %uynwsqns
%-(dZ+P+6c/lH-==a $eTl/i - tdz + P + %M -3 = auop vM
z ayl u! mop yoM ou si alay ‘ze luamaaour B 103 uog~o 30 aug pm apn@kur s)f sa%ueyD H q%noqlpt asnmaq H %U!AIOAU!maI ou sy aJayJ *s!sA~eut!cy.usulCp ayl IOU pur! s8upds aql 30 suo!gagap 5gaJs ayj paJs Quo Aaql asnmaq paJo&! aq ~03 ‘,A asw S!Q~ut ‘swa) JUOJSUOL) ‘(ye
-
ye
-
“146-
?)s3 -
,_,g = df3 -
,A = A = =a
*awuploo~ z aql 103 ~~03 jmla)xa aql s! Za aJayM ze A = ze ‘Z, = auop ylohi
alswploo9
ze watuamIds!p t? .IO~ ‘SYXaXo3 aQ$QXQM ut )ey) se uo~~~a.upsum aQJ u! s! $uatuamIds!p ayl uayM uralsds ayl uo saDlo pmalxa aq$ Aq auop SI ylonn ang!sod $VQ$ aJoN *‘be luaurabow IIwus t! 103 _,j a~03 Izg.IaA ayl pu?? H dq auop 1.10~ aQl%uyywIe~ 1(q pun03 6I!sEa ~sow a.w S~JOJ Ivula)xa aQJ
aDlo Itz~uoz~.xo~ay)
252
TRACTOR
2.3.
RIDE
VIBRATION
Calculation of the natural frequencies (eigenvalues) and mode shapes (eigenvectors)
The four previously derived equations can be written in matrix form as [A]ii+[B]i+[c]u==o where the vector u represents the coordinates of the system, in this case, z, 4, 0,, and 8,. [A] is the “mass” matrix, [B] the “damping” matrix and [C] the “displacement” matrix. The computer program used to solve this problem requires 1st order equations. So the four 2nd order equations must be written as eight 1st order. A new set of subsidiary variables must therefore be defined.
&(“,= il. The system equation can now be written d
Ml -&(9) + PI 6) + [Cl (u) = cl
v=[I z. ,
WI$00 - VI (3 = 0
where Z is the unit matrix. The equations now only involve u and ir so that by defining a variable they can be described by one equation.
[A’1$(v) + (B’)(v)= 0 (A’)and (B’) are 8 x 8 matrices and v in this particular case=
The computer program requires the matrices [A’] and [B’] as input and they are derived from the tractor and implement problem in Appendix B. 2.4. Results Table I is a summary of the natural frequencies and percentage critical damping values for various tractor, plough and soil force combinations. The natural frequency is the imaginary part of the eigenvalue calculated by the computer programme and the percentage critical damping is approximately the real part of the eigenvalue divided by the imaginary part. The left-hand column indicates the coordinate which had the greatest displacement amplitude at that frequency. At a particular frequency, vibration occurs in all coordinate directions and if the vibration amplitudes in two directions are similar, coupling is said to occur between those coordinates. Where significant coupling occurred an arrow marks the other coordinates involved. This information is found from the eigenvectors. The most important result in this table is the change in damping and natural frequency of tractor pitch between the tractor alone, and when it is ploughing. The percentage critical damping increases from 0.04 to O-67, which indicates that the vertical soil forces on the implement are
253
D. A. CROLLA
TABLEI Natural frequencies (Hz) and approximate percentage critical damping (in brackets) of various tractor, plough and soil force combinations. The arrows indicate significant coupling between coordinates
-
-
Tractor alone
--
--____-2.8
(0.07)
4.0
(0.04)
Tractor with plough rigidly attached
Plough alone
2.9 1.8
Plough alone with soil forces acting
_.
(0.12),
2.9
1 (0.04) ’ -
Tractor and ptough with only draught force acting
Tractor and plough with soil forces acting
(0.04)
2.8
(0.04)
2.8
(0.05)
2.8
(0.05)
(0.04)
1.7
(0.04)
1.6
(0.04)
1.4
(0.67)
6.9
(0.17)^
6.9
(0.17)^
6.7
(0.20);
3.6
(0.08)’
6.1
6.4
Tractor with plough raised on linkage --
(0.16)*
(0.43)’’ 3.7
2.7
(0.03)‘1
3.7
(0.03)‘J
,
’
-
exerting a high damping force on tractor pitch motion and the natural frequency decreases from 4.0 to l-4 Hz. Thus tractor ride vibration is significantly different when ploughing from when the tractor is alone. The plough is also affected when attached to the tractor. Thus can be seen by comparing the results for the plough alone with those when the tractor is ploughing. The natural frequencies of the plough and linkage alone assuming soil forces are acting are 6.4 and 2.9 Hz, with damping ratios of O-12and 0.04. When the tractor is ploughing, these become 6.7 and 3.6 Hz with damping ratios of 0.20 and 0.08.
03-
0.1
T
c*
Only one result ‘or ploughing at this speed
0 0
2
1
Forward
Fig. 3. Vertical vibration levels measured
speed
3(m/s)
at the driver’s seat for various form operations2
254
TRACTOR
z 5 t >
0.6.
*Only one result for 2 furrow plough \
0
1
2
3
4
5
RIDE
VIBRATION
J 6
Number of furrows
Fig. 4. Decrease
in vertical vibration level with increasing plough size (averaged over speed range l-2.5
rn/s)l
So both tractor and plough behaviour are altered when they are combined, i.e. when the tractor is ploughing. This has two important effects. Firstly tractor ride vibration will be lower when ploughing because tractor pitch and bounce are damped by the vertical force on the plough. Secondly, the damping forces are transmitted by the tractor linkage and since the linkage forces are sensed by the draught control, it will receive spurious signals which are not due to changes in implement draught. Also, for vertical motion, the human body is most sensitive to frequencies of 4 to 8 Hz. Vertical motion has components due to both pitch and bounce and the pitch component for the tractor alone occurs at 4 Hz whereas when ploughing it occurs at l-4 Hz where the body is slightly more tolerant. 3. Field results A recent survey* of driver vibration levels in 71 tractors engaged in a variety of field work indicated that ride levels when ploughing were lower than for other operations (Fig. 3). In this work, the N.I.A.E. ridemeter was used to measure the acceleration levels at the driver’s seat. This instrument is frequency-weighted so that its vibration response characteristic closely matches the response of the human body and it provides an average normalized ride level over a test period from 20 s to 5 min. This survey also indicated that there was some correlation between ride level and plough size. When the ride level measurements of 15 tractors which were ploughing were plotted against implement size, there was a trend of decreasing ride vibration level with increasing implement size (Fig. 4). This would be expected because the vertical damping force on the implement is proportional to draught force, and also the line of action of the force acts further away from the tractor and its movement arm increases as implement size increases. TABLE II Fields used
for ride meamementa when ph@ing
I Field number
I Soil type
1 2 3
Clay loam Clay loam Sandy loam
4
Sandy loam
I Suyface
Stubble Stubble Stubble, disced 5 cm deep, limed Stubble
Approximate soil resistance, N/ma (Ibf/irP)
83 ooo 83 000 41 ooo
(12) (12) (6)
55 ooo
(8)
D.
A.
255
CROLLA
?I
E
/
a,
/
l.O-
/
4
s
3
Tractor
alone
(in furrow)
/
OB-
;
e .f 1
0.6-
r P 0.4-
0.2 t
O._ 0
1 Forward
2 speed (m/s)
Fig. 5. Vertical vibration levels measured in field 1
01 0
1 Forward
2 speed
(m/s1
Fig. 6. Vertical vibration levels measured in field 2
256
TRACTOR
RIDE
VIBRATION
1.6
Ploughing
: I LT ‘5 : l.O5? > a y k >
I / x
I’ I’
VB/ /i 0.6 1
3
2
speed (m/s)
Forward
Fig. 7. Vertical vibration levels measured in field 3
Since these results were measured over a very wide range of tractors, field surfaces, implements and tractors drivers, they could not be used to draw conclusive evidence for a particular case, such as the decreased ride level when ploughing. Further fieldwork was necessary to investigate the effects of ploughing on tractor ride levels. Four fields were used for further measurements (Table II). Acceleration levels at the driver’s seat in the vertical, lateral and horizontal directions were measured with the N.I.A.E. ridemeter. In each field, ride level measurements were made with the tractor ploughing and the tractor alone in the furrow at various speeds. The test period for one measurement varied depending on the field width but for each speed at least six measurements were taken and averaged. All measurements were made with a medium power tractor (63 kW p.t.o.) and three furrow plough. In field 3, measurements were also made with a five furrow plough, both fully mounted and with some weight carried on the depth wheel. The field measurements made were of the acceleration levels of the driver measured at his seat, and therefore included the effect of the seat suspension. The theoretical analysis did not include the seat suspension but this is not considered to be important for this work since no attempt
s-o--
rC%ctor
,
,
/’
PC
/x--
alone (in furrow)
Ploughing
0.4,; 3 ForwoO d speed
(m/s)
Fig. 8. Vertical vibration levels measured in field 4
1
D. A.
257
CROLLA
is being made to predict the actual ride vibration levels which would certainly require information about the seat suspension and also the ground surface. The trends of the effect of the plough on tractor vibration are the important aspects of this work so the inclusion of the seat suspension in the theoretical analysis is not really necessary. Care was however taken to ensure that the seat suspension was not changed during the experiments and the same tractor driver was used throughout. Tractor vertical vibration levels for fields 1 and 2 are plotted against forward speed in Figs 5 and 6. The measurements with the tractor alone are at a slightly higher speed than when ploughing because the wheels are not slipping. The results for the two fields are very similar, and at 2.2 m/s (5 mile/h), vertical vibration level is decreased by about 507, due to the damping effect of the plough. Fig. 7 is a graph of vertical vibration level against speed for field 3 where ploughing speeds up to 2.8 m/s (6.3 mile/h) were used. The ride level for the tractor alone shows a very different trend to fields 1 and 2, where the ride level became increasingly high as speed increased. Above about 2.5 m/s (5.5 mile/h), the curve flattens out and ride level remains fairly constant up to 3.1 m/s (7 mile/h). The ride level when ploughing is consistently lower, about 25% lower at 2.5 m/s (5.5 mile/h) for example, but the curves look as if they will coincide at about 3.1 m/s. The reason for the flattening off of the ride level may be that the surface was fairly loose and deformable because it had been disced. As the tractor crossed the surface, the soil deformation provided a certain amount of damping which had more effect on ride level at higher speed. 1.2
CT f
Ploughing (3 furrows) 1.0. Ploughing (5 furrow ,,*plough with depth
i _x_.x-
2 Forward
Fig. 9. Field 3; efect
Ploughing (5 furrows)
3 speed (m/s)
of different plough size and addition of depth wheel on vertical
vibration
The results for field 4, shown in Fig. 8, show a similar trend. Although the surface had not been disced, it was fairly soft and deformable and the tractor left obvious wheel ruts where it had travelled. Vertical vibration level in field 4 was consistently above 20% lower when ploughing over the speed range from 1.8 to 2.8 m/s (4 to 6.3 mile/h). In field 3, measurements were made using a five furrow plough, and the results are compared with those for a three furrow plough in Fig. 9. At 2.2 m/s rrde vibration is about 30% lower with the five furrow plough than with the three furrow plough. This is because the vertical damping force on the plough is proportional to draught force, which is greater with the five furrow plough and also its line of action is further back and hence the moment acting on the tractor is greater. The measurements made with the fully mounted five furrow plough were compared with those made when a depth wheel was attached to the rear of the plough to assist depth control. Some of the plough weight was carried by the wheel and hence not all the vertical force on the plough
258
TRACTOR
RIDE
VIBRATION
acted on the tractor. This would be expected to reduce the damping effect of the plough, and the results in Fig. 9 show that it did. The ride level with the fully mounted plough was about 25 % lower than when the plough was fitted with a depth wheel at the rear. So far only vibration in the vertical direction has been discussed. In the longitudinal and lateral directions, no conclusive differences in ride level were found between the tractor alone and when ploughing. In fields 2 and 4 there was some evidence that the longitudinal vibration might be reduced slightly at the highest speeds (Fig. IO), but more results and preferably at even higher speeds would be necessary to confirm this. Longitudinal vibration may be expected to be reduced because the plough damps tractor pitch motion and, since the driver is not seated at the centre of pitch, his motion in the longitudinal direction should be affected. Some measurements were made with the tractor running on the land out of the furrow and without a plough. The ride levels in all three directions tended to be higher than when the tractor was in the furrow. Not enough results were taken to be conclusive but this trend might be expected since in the furrow two of the tractor wheels travel on a smoother surface than the land. Tractor
2 Forward
alone
3 speed
(m/s)
Fig. IO. Longitudinal acceleraion levels in fields 2 and 4
4.
Discussion
The results obtained in this work have several implications for future tractor design because both main conclusions become increasingly important at ploughing speeds above about 2.2 m/s (5 mile/h). Tractor ride vibration levels are significantly reduced when ploughing and because the damping forces are transmitted by the tractor linkage, draught controls which sense linkage force are less effective. Considering the amount of work that has been carried out into tractor ride dynamics, it is perhaps surprising that to date there has been no published work which includes the effect of implements or trailers. This work has shown that a plough significantly affects tractor behaviour and it is likely that other implements will also modify tractor behaviour though perhaps not to such an extent. Cultivators for example have similar soil force characteristics to ploughs except that the first term in the vertical force relationship which was assumed constant for the plough, is significant. Steady state vertical force increases with depth, so it opposes the damping term in the vertical force equation. In fact, if it is greater than the second term, negative damping would result and the system would be unstable. The damping term is proportional to ((draught force)/(forward speed) x rate of change of depth) so, as forward speed increases, the damping term decreases and stability decreases. The draught control response also affects stability, because when the linkage is moved, the vertical force changes, due to the rate of change of depth term. It is possible therefore for the draught control response to excite a natural frequency of the tractor and implement and increase tractor ride vibration.
D. A.
259
CROLLA
In general, any implement which is mounted on the tractor linkage will alter the tractor natural frequencies especially in pitch, because the implement weight acts well behind the tractor centre of gravity. Two-wheel trailers which transfer weight on to the tractor drawbar would also be expected to affect tractor ride significantly. There is scope for further work to investigate the effects of implements and trailers on tractor behaviour. The future of draught controls which sense linkage force must be questioned at high speed. Future trends in implement control will be influenced by the advances in cultivation implement design and in improvements to tractor ride at higher speeds. The benefits of improved ride level for high speed ploughing may be an important advantage o\er other cultivating techniques. The plough would have to be mounted and an improved draught control would be required since the performance of present linkage force sensing controls has been shown to be inadequate5 over about 2.2 m/s (5 mile/h). The force fluctuations due to implement damping are in the same frequency range as those due to implement depth variations, so they cannot be filtered out. A possible alternative is a control which senses draught force only, driveline torque, engine speed or wheelslip. A pure draught sensing control could be arranged without too much modification to present designs, by having two sensing units, one sensing top link force and the other sensing lower link force. The addition of the two signals, eliminates the vertical force and the resulting sum is equal to the draught force only. 5. Conclusions A theoretical model showed that when ploughing, the plough exerted considerable damping on the tractor pitch. Fieldwork confirmed this and tractor vertical ride vibration level measured at the driver’s seat was reduced by up to 50% when ploughing heavy land and 30% on light land. The significant effect of the plough on tractor behaviour indicates that work on tractor ride dynamics should be extended to include implements and trailers. Also the damping forces transmitted from the plough to the tractor cause fluctuation% in linkage force, and draught controls which sense this force receive spurious signals making them inadequate at speeds above about 2.2 m/s (5 mile/h). REFERENCES
Dwyer, M. J. The braking performance of tractor-trailer combinations. J. agric. Engng Res., 1970 15 149 Stayner, R. M.; Bean, A. G. M. Tractor Ride Investigations; a survey of vibrations experienced by drivers in fieldwork.
N.I.A.E. Departmental Note DN/E/578/1445 (unpublished), 1975
Dwyer, M. J. The dynamic performance of tractor-implement combinations. Proc. Instn Mech. Engrs, 1969-70 184 Part 3Q Crolla, D. A. A computer programme for simulating the field performance of a tractor and mounted cultivation implements under varying load condition. N.I.A.E. Note 62/1410 (unpublished), 1975 Crolla, D. A.; Pearson, G. Response of tractor draught controls to nmdom inputs. J. agric. Engng Res., 1975 20 181.
Hilton, D. J. A frequency weighted ride-meter. J. agric. Engng Res., 1970 15 379 Appendix A Values of parameters
The figures used are from a ballasted 63 p.t.o. kW tractor and three furrow plough. Tractor mass M 4300 kg J Tractor moment of inertia 2940 kg m2 Plough mass 545 kg Plough moment of inertia 727 kg m2 5 Front tyre stiffness 300 000 N/m k, Front tyre damping 725 N s/m Cf Rear tyre stiffness 400 000 N/m k,
260
TRACTOR
Rear tyre damping Hydraulic stiffness Hydraulic damping Top link sensing unit stiffness Top link sensing unit damping Distances (see Fig. 1)
Equivalent soil stiffness Equivalent soil damping
RIDE
3500 N s/m 1400 000 N/m 14 000 N s/m 2 500 000 N/m 0 lm lm O-5 m 0.79 m 1.43 m 0.27 m O-76 m 0.23 m 43 700 N/m 8700 N s/m
Appendix B The four second-order equations of motion can be written a,i’+a,~+a,B,+a,B,+a,i+a,~+a,B,+a,B,+a,z+a,,~=O blP+b,~+b,B;~+b,~,+b,i+b,~+b,8~+b~~~+bs~+b~o~+b~~~~+b~,~z=O c~i-+c,~+c,B,+c,ii,+c,i+c,~+C,B~+C*~2+CgZi-C~0~2=0 d,i+d,d;+d,B,+d,B,+d,i+d,~+d,81+ds82+d,z+d,,~+d,,B,+d,,8,=0
where
a,=M+m -ml, as=-
-ml, Cl=-
-ml1 2 -ml, a4=2 a5=cf+cr+c, a, = bc,--ac,-c,14 a, =-c,ll a8=-cslz a,=kf+k, aIo= bk,--ck, -ml, b,=2
as=-
b,=J+Ie+m1,2 b,=ml,l, b,=ml,l, b,=bc,-ac,+csll b,=b2cf+a2c,+cs1,2 b, = c,l,l, b8=cs121, b,=bk.,-ak,+k,(d+h, b,,=b2kf+a2k,-k,14 b,,=-ksl, (2d+h,) b,,=-k,l, (Tdfh,)
2
c,=ml,l, c,=ml12 c,=ml,12 cg=-c 1 cs = c,r,;t Cll12 c,=+c,112 4 r,=c,l,12
2
cg=-
kA2 4
+-q
(2d+h,)
d,=ml,l, d,=ml,l, d,=lo2+m122 d,=--c,l, d,=c,l,l, d,=c,l,l, d8=c2132+cs122 ds=k,(d+h,) 4o=-k,(dr,+dl,+h,l,) d,,=-k,l,(d+h,) d12=k,l,“-k,1,(2d+h,)
VIBRATION
D. A.
CROLLA
261
Combining these four equations with the four first-order equations gives the matrix equation
CA’$@I 1 + [B’l(v) = 0 where
y=
-2
--
k 0:
i -s:_ A’ E-
B' =
a, a, a3 a4 0 0 0 0 b, b, b, 6, 0 0 0 0 c,o 0 0 0 ~~~d,OOOO 00001000 00000100 00000010 _00000001 a5 a6 a, a, a, a,,0 0 b, b, b, b, b, blob,, b,, 0 yyj72dd>d 00 _; o6 o7 o8 o9 olooll 012 o-1 0 0 0 o-1 0 -0 0 o-1
0 0 0
0 0 0
0 0 0
0 0 0
Effect of Cultivation Implements on Tractor Ride Vibration and Implications for Implement Control D. A. CROLLA* The effect of implements on tractor ride vibration is not well understood. Nor is the effect of tractor ride vibration on control of the implement. Previous work on tractor ride dynamics has concentrated on motion of the tractor alone. A four degree of freedom model of a medium power tractor and three furrow plough with soil forces acting on it was developed. The model indicated that the plough exerted considerable damping on the tractor pitch and bounce. As forward speed and tractor vibration levels increased, the damping was likely to become more important. Field results confirmed these predictions. Tractor vertical ride vibration levels, measured with the N.I.A.E. ridemeter were reduced by up to 50% in heavy land and 30% in light land. A five furrow plough reduced tractor ride vibration level in the vertical direction more than a three furrow plough, because the soil forces acting on it were greater. These results have two important implications. Attempts to predict tractor ride vibration should include the effect of the implement or trailer. The damping forces are transmitted to the tractor by the tractor linkage and since conventional draught controls sense linkage force, they receive spurious signals due to the damping forces. These force fluctuations are in the same frequency range as those due to changes in implement depth and cannot therefore be filtered out. The problem increases with speed and if high ploughing speeds i.e. above 2-2 m/s (5 mile/h) are used in the future, an alternative to conventional linkage force sensing controls must be provided.
1. Introduction Previous work on tractor dynamics has considered the tractor alone in attempts to investigate its ride characteristics. This is perhaps surprising since tractors are operated with either an
implement or trailer for most of the time. Some work has been carried out into the behaviour of a tractor and trailer under braking’ but otherwise the effects of implements and trailers on the tractor have been neglected. Recently a survey* of tractor vibration levels for a variety of farm tasks has been carried out and although significant differences were found for various operations, a trend of increasing vibration levels with speed was established. The effect of the implement and draught control response on tractor dynamic performance in the longitudinal direction has been studied3 and a computer model has been devised to simulate the field performance of a tractor operating cultivation implements4. Tractor dynamic motion in the vertical plane however is ignored, tractor vertical displacement being assumed to equal the displacement of the surface contour. This assumption can be questioned at cultivating speeds above 2.2 m/s and it has been suggested that it may result in predicted draught control performance being slightly better than measured performances above 2.2 m/s. If tractor pitch and bounce accelerations are significant their effect will be to increase implement depth variation or if the soil forces prevent depth variation to increase linkage forces and cause a deterioration in draught control performance. Both these effects result in an increased draught variation. In this paper, the equations of motion for a tractor and implement are derived using Lagrange’s equation. The natural frequencies (eigenvalues) and damping ratios are calculated for a medium power tractor with and without a plough. Field measurements, carried out to validate the predicted results are also described. ??N.I.A.E., Wrest Park, Silsoe, Bedford Received 25 July 1975; accepted in t-wised form 23 December 1975
241
248
TRACTOR
RIDE
VIBRATION
LIST OF SYMBOLS a b
distance from rear wheel to tractor c.g. distance from front wheel to tractor
i&ping coefficient of hydraulic lift damping coefficient of sensing unit c2 damping coefficient of front tyre Cf damping coefficient of rear tyre G equivalent damping coefficient of c, soil d implement depth height of tractor c.g. above ground h, hi? height of lower links above ground. H draught force ICCl inertia of implement about its c.g. inertia of implement about centre of 1, gravity of tractor I a2 inertia of implement about lower link ends inertia of tractor about its c.g. J stiffness of hydraulic lift kl stiffness of sensing unit k, stiffness of front tyre k.t stiffness of rear tyre k, Cl
2.
k,
rate of change of draught force with depth II distance from rear wheel to lower link ends I, distance from lower link ends to plough c.g. mast height 1, 14 (a+ll+&) m implement mass M tractor mass ith generalized coordinate 41 4 I external force for particular qr Q coordinate V vertical force on the plough plough velocity VP tractor forward displacement x x, plough forward displacement tractor vertical displacement z plough vertical displacement tractor pitch ! rotation of lower links (relative to 1 tractor) 62 rotation of implement (relative to lower link ends)
Theoretical analysis
2.1. The tractor/implement model Fig. 2 is a diagram of the tractor and implement with the external soil forces acting on it. The four degrees of freedom, or generalized coordinates are, z-tractor bounce, +-tractor pitch, &-rotation of lower links and f3,-rotation of implement. The tractor tyres, compliance in the hydraulic lift and top link sensing unit are all modelled by equivalent springs and dampers. Also the following assumptions are made. (1) The top and lower links are parallel. (2) The lift rods are perpendicular to the lower links. (3) The links have negligible mass compared with that of the tractor or implement. (4) The spring kl is always in tension because the vertical force on the plough is always positive and is therefore considered to be double-acting, as is the damper cl. If the vertical force on the plough was not always positive then the assumption would be invalid because tractor linkages are not restrained in the upward direction. There are two external soil forces acting on the implement, the draught force N and the vertical force V. H is a function of implement depth for a particular field and can be assumed to be proportional to depth over a small range. H=k,(d+z,).
D.
A.
249
CROLLA
V has been found previously3 to be a function of implement depth, rate of change of depth and H Y=f(d+z,)-I-z%. P
The first term,f(d+z,) does not change significantly for small changes in depth, z,, and can be approximated to a constant term V’=f(d). Also, assuming that H remains constant at its value at the mean depth, d, V can be approximated to
and for a constant forward speed v =IV’ - c,i,. The figures used in this analysis refer to a medium power tractor and three furrow mouldboard plough and values of all parameters used are given in Appendix A. DArrows indicate coordinates
genemliscd
b
Fig. 1. Tractor and implement showinggeneralised coordinates and forces (nor to scale)
2.2. Derivation of the equations of motion The method of Lagrange is used to obtain the differential equation of motion of the tractor and implement. The problem must first be expressed in generalized coordinates qt. These are independent of one another and i is the minimum number of them necessary to specify any part of the system. Lagrange’s equation is a differentiation of the energy of the system written in the generalized coordinates . . . (1)
where
= kinetic energy, U = potential energy, K = dissipative energy and Qq t = generalized external force for the q i coordinate. T
This method has advantages for systems with multiple degrees of freedom because only the system energies need to be found. The forces of constraint which do no work but which often complicate the force-mass-acceleration equations are excluded. Also, the calculation of energies
250
TRACTOR
RIDE VIBRATION
requires only the velocities and not the accelerations of the system components which are often more difficult to compute. Substituting each of the generalized coordinates, z, c#, & and 8, into Eqn (1) gives one equation of motion of the system. To calculate the kinetic (T), potential (U) and dissipative (K) energies we need the velocities of the main components. The tractor velocity is simple but the implement velocity, vp, is more complicated because it involves all the generalized coordinates. The vector diagram for the velocity components due to each coordinate is shown in Fig. 2. Hence,
v,2 = i,2 + jcp2 + &cos+-~~ = @,I, COSB,+ ~,i2co~e2 + (&sin e1 + b,l, sin e2 + &, sin$)2
which becomes VP2= &1,2 + 822122 + C$2142 + 2r,1,e,e2 + 21,1,&4 + 21,1,e,Cj+ i2 - i(B,Z, + e2r2+ &) assuming that cos (0, - e,), cos (0, - 4) and cos (0, - $) all equal unity because the angular displacements are small.
Fig. 2.Vector diagram to calculate plough velocity
The energies of the system can now be calculated. Kinetic energy :
T = &Mi2 + iJ& + $mv,” + &
Potential energy :
u = pi&
Dissipative energy :
K = &c,(i + b&” + $cr(.i +
&
+ b#
(&4)2
2
+
+ tJw
-
4;’f &di
q&”
a$@
ic2(~2Q2.
Before substituting these into the Lagrange Eqn (1) the external force Qql for each coordinate must be known.
‘pa)t3In3Im a~ ut?3 (sJopaAua%!a) sadeys apotu pue (sanIsAua@a) sapuanba;r3
aq$ ‘Jauuatu sg
u! uor~oux 30 suoyymba leyIaJa&JgI rapJo-puoDas mo3 aq$ paugqo
Ismltm %U!AITH
.o=~(‘ytl--~q)+z(‘y+~~)+ “yj”3 - ‘9’133 - f cpl”3- ‘3v - Jq) + q (“3+ ‘3 + 4) +
*Molaq ua@ SFQuo uoya.up z ayl.103 aldumxa ut? ‘ss%old @%a~ Jay)tzr B si S%J cmns *wa)sds aql30 uogotu 30 suoymba mo3 a@ 01 awuploo:, pazgelaua% yma 103 (I) ubg II! pa$nl!$sqns aq MOU ku “ba pm x ‘fl ‘J ~03 suoyssaldxa aqJ
‘W9 - ‘1’fI- “19- ZIZlf3+ (“1”e- ‘1%- “I$ - ZX”Y + PZ)“Y- = ““d “se (“1~- Vz + p + %M -3 = =-w voM zee luawaDt?Ids!p B .103puo
tee ~uaumqds!p *Cize- Ye - “if - W3
t? ~03 s!sAp~~a.~e~!tu!sE %upn
+ (zize - IlIe - % - 3 (“Y+ PZYY -
= %
‘srnlaj laplo-puo9as os~e pm alo3aq se suna) ~uersuo:, Bu!~ou81
-ydg3 - ,A) - tdz + p + “?I)(“z + pyy -
= pa /i pm
saA!z
H ~03 %uynwsqns
%-(dZ+P+6c/lH-==a $eTl/i - tdz + P + %M -3 = auop vM
z ayl u! mop yoM ou si alay ‘ze luamaaour B 103 uog~o 30 aug pm apn@kur s)f sa%ueyD H q%noqlpt asnmaq H %U!AIOAU!maI ou sy aJayJ *s!sA~eut!cy.usulCp ayl IOU pur! s8upds aql 30 suo!gagap 5gaJs ayj paJs Quo Aaql asnmaq paJo&! aq ~03 ‘,A asw S!Q~ut ‘swa) JUOJSUOL) ‘(ye
-
ye
-
“146-
?)s3 -
,_,g = df3 -
,A = A = =a
*awuploo~ z aql 103 ~~03 jmla)xa aql s! Za aJayM ze A = ze ‘Z, = auop ylohi
alswploo9
ze watuamIds!p t? .IO~ ‘SYXaXo3 aQ$QXQM ut )ey) se uo~~~a.upsum aQJ u! s! $uatuamIds!p ayl uayM uralsds ayl uo saDlo pmalxa aq$ Aq auop SI ylonn ang!sod $VQ$ aJoN *‘be luaurabow IIwus t! 103 _,j a~03 Izg.IaA ayl pu?? H dq auop 1.10~ aQl%uyywIe~ 1(q pun03 6I!sEa ~sow a.w S~JOJ Ivula)xa aQJ
aDlo Itz~uoz~.xo~ay)
252
TRACTOR
2.3.
RIDE
VIBRATION
Calculation of the natural frequencies (eigenvalues) and mode shapes (eigenvectors)
The four previously derived equations can be written in matrix form as [A]ii+[B]i+[c]u==o where the vector u represents the coordinates of the system, in this case, z, 4, 0,, and 8,. [A] is the “mass” matrix, [B] the “damping” matrix and [C] the “displacement” matrix. The computer program used to solve this problem requires 1st order equations. So the four 2nd order equations must be written as eight 1st order. A new set of subsidiary variables must therefore be defined.
&(“,= il. The system equation can now be written d
Ml -&(9) + PI 6) + [Cl (u) = cl
v=[I z. ,
WI$00 - VI (3 = 0
where Z is the unit matrix. The equations now only involve u and ir so that by defining a variable they can be described by one equation.
[A’1$(v) + (B’)(v)= 0 (A’)and (B’) are 8 x 8 matrices and v in this particular case=
The computer program requires the matrices [A’] and [B’] as input and they are derived from the tractor and implement problem in Appendix B. 2.4. Results Table I is a summary of the natural frequencies and percentage critical damping values for various tractor, plough and soil force combinations. The natural frequency is the imaginary part of the eigenvalue calculated by the computer programme and the percentage critical damping is approximately the real part of the eigenvalue divided by the imaginary part. The left-hand column indicates the coordinate which had the greatest displacement amplitude at that frequency. At a particular frequency, vibration occurs in all coordinate directions and if the vibration amplitudes in two directions are similar, coupling is said to occur between those coordinates. Where significant coupling occurred an arrow marks the other coordinates involved. This information is found from the eigenvectors. The most important result in this table is the change in damping and natural frequency of tractor pitch between the tractor alone, and when it is ploughing. The percentage critical damping increases from 0.04 to O-67, which indicates that the vertical soil forces on the implement are
253
D. A. CROLLA
TABLEI Natural frequencies (Hz) and approximate percentage critical damping (in brackets) of various tractor, plough and soil force combinations. The arrows indicate significant coupling between coordinates
-
-
Tractor alone
--
--____-2.8
(0.07)
4.0
(0.04)
Tractor with plough rigidly attached
Plough alone
2.9 1.8
Plough alone with soil forces acting
_.
(0.12),
2.9
1 (0.04) ’ -
Tractor and ptough with only draught force acting
Tractor and plough with soil forces acting
(0.04)
2.8
(0.04)
2.8
(0.05)
2.8
(0.05)
(0.04)
1.7
(0.04)
1.6
(0.04)
1.4
(0.67)
6.9
(0.17)^
6.9
(0.17)^
6.7
(0.20);
3.6
(0.08)’
6.1
6.4
Tractor with plough raised on linkage --
(0.16)*
(0.43)’’ 3.7
2.7
(0.03)‘1
3.7
(0.03)‘J
,
’
-
exerting a high damping force on tractor pitch motion and the natural frequency decreases from 4.0 to l-4 Hz. Thus tractor ride vibration is significantly different when ploughing from when the tractor is alone. The plough is also affected when attached to the tractor. Thus can be seen by comparing the results for the plough alone with those when the tractor is ploughing. The natural frequencies of the plough and linkage alone assuming soil forces are acting are 6.4 and 2.9 Hz, with damping ratios of O-12and 0.04. When the tractor is ploughing, these become 6.7 and 3.6 Hz with damping ratios of 0.20 and 0.08.
03-
0.1
T
c*
Only one result ‘or ploughing at this speed
0 0
2
1
Forward
Fig. 3. Vertical vibration levels measured
speed
3(m/s)
at the driver’s seat for various form operations2
254
TRACTOR
z 5 t >
0.6.
*Only one result for 2 furrow plough \
0
1
2
3
4
5
RIDE
VIBRATION
J 6
Number of furrows
Fig. 4. Decrease
in vertical vibration level with increasing plough size (averaged over speed range l-2.5
rn/s)l
So both tractor and plough behaviour are altered when they are combined, i.e. when the tractor is ploughing. This has two important effects. Firstly tractor ride vibration will be lower when ploughing because tractor pitch and bounce are damped by the vertical force on the plough. Secondly, the damping forces are transmitted by the tractor linkage and since the linkage forces are sensed by the draught control, it will receive spurious signals which are not due to changes in implement draught. Also, for vertical motion, the human body is most sensitive to frequencies of 4 to 8 Hz. Vertical motion has components due to both pitch and bounce and the pitch component for the tractor alone occurs at 4 Hz whereas when ploughing it occurs at l-4 Hz where the body is slightly more tolerant. 3. Field results A recent survey* of driver vibration levels in 71 tractors engaged in a variety of field work indicated that ride levels when ploughing were lower than for other operations (Fig. 3). In this work, the N.I.A.E. ridemeter was used to measure the acceleration levels at the driver’s seat. This instrument is frequency-weighted so that its vibration response characteristic closely matches the response of the human body and it provides an average normalized ride level over a test period from 20 s to 5 min. This survey also indicated that there was some correlation between ride level and plough size. When the ride level measurements of 15 tractors which were ploughing were plotted against implement size, there was a trend of decreasing ride vibration level with increasing implement size (Fig. 4). This would be expected because the vertical damping force on the implement is proportional to draught force, and also the line of action of the force acts further away from the tractor and its movement arm increases as implement size increases. TABLE II Fields used
for ride meamementa when ph@ing
I Field number
I Soil type
1 2 3
Clay loam Clay loam Sandy loam
4
Sandy loam
I Suyface
Stubble Stubble Stubble, disced 5 cm deep, limed Stubble
Approximate soil resistance, N/ma (Ibf/irP)
83 ooo 83 000 41 ooo
(12) (12) (6)
55 ooo
(8)
D.
A.
255
CROLLA
?I
E
/
a,
/
l.O-
/
4
s
3
Tractor
alone
(in furrow)
/
OB-
;
e .f 1
0.6-
r P 0.4-
0.2 t
O._ 0
1 Forward
2 speed (m/s)
Fig. 5. Vertical vibration levels measured in field 1
01 0
1 Forward
2 speed
(m/s1
Fig. 6. Vertical vibration levels measured in field 2
256
TRACTOR
RIDE
VIBRATION
1.6
Ploughing
: I LT ‘5 : l.O5? > a y k >
I / x
I’ I’
VB/ /i 0.6 1
3
2
speed (m/s)
Forward
Fig. 7. Vertical vibration levels measured in field 3
Since these results were measured over a very wide range of tractors, field surfaces, implements and tractors drivers, they could not be used to draw conclusive evidence for a particular case, such as the decreased ride level when ploughing. Further fieldwork was necessary to investigate the effects of ploughing on tractor ride levels. Four fields were used for further measurements (Table II). Acceleration levels at the driver’s seat in the vertical, lateral and horizontal directions were measured with the N.I.A.E. ridemeter. In each field, ride level measurements were made with the tractor ploughing and the tractor alone in the furrow at various speeds. The test period for one measurement varied depending on the field width but for each speed at least six measurements were taken and averaged. All measurements were made with a medium power tractor (63 kW p.t.o.) and three furrow plough. In field 3, measurements were also made with a five furrow plough, both fully mounted and with some weight carried on the depth wheel. The field measurements made were of the acceleration levels of the driver measured at his seat, and therefore included the effect of the seat suspension. The theoretical analysis did not include the seat suspension but this is not considered to be important for this work since no attempt
s-o--
rC%ctor
,
,
/’
PC
/x--
alone (in furrow)
Ploughing
0.4,; 3 ForwoO d speed
(m/s)
Fig. 8. Vertical vibration levels measured in field 4
1
D. A.
257
CROLLA
is being made to predict the actual ride vibration levels which would certainly require information about the seat suspension and also the ground surface. The trends of the effect of the plough on tractor vibration are the important aspects of this work so the inclusion of the seat suspension in the theoretical analysis is not really necessary. Care was however taken to ensure that the seat suspension was not changed during the experiments and the same tractor driver was used throughout. Tractor vertical vibration levels for fields 1 and 2 are plotted against forward speed in Figs 5 and 6. The measurements with the tractor alone are at a slightly higher speed than when ploughing because the wheels are not slipping. The results for the two fields are very similar, and at 2.2 m/s (5 mile/h), vertical vibration level is decreased by about 507, due to the damping effect of the plough. Fig. 7 is a graph of vertical vibration level against speed for field 3 where ploughing speeds up to 2.8 m/s (6.3 mile/h) were used. The ride level for the tractor alone shows a very different trend to fields 1 and 2, where the ride level became increasingly high as speed increased. Above about 2.5 m/s (5.5 mile/h), the curve flattens out and ride level remains fairly constant up to 3.1 m/s (7 mile/h). The ride level when ploughing is consistently lower, about 25% lower at 2.5 m/s (5.5 mile/h) for example, but the curves look as if they will coincide at about 3.1 m/s. The reason for the flattening off of the ride level may be that the surface was fairly loose and deformable because it had been disced. As the tractor crossed the surface, the soil deformation provided a certain amount of damping which had more effect on ride level at higher speed. 1.2
CT f
Ploughing (3 furrows) 1.0. Ploughing (5 furrow ,,*plough with depth
i _x_.x-
2 Forward
Fig. 9. Field 3; efect
Ploughing (5 furrows)
3 speed (m/s)
of different plough size and addition of depth wheel on vertical
vibration
The results for field 4, shown in Fig. 8, show a similar trend. Although the surface had not been disced, it was fairly soft and deformable and the tractor left obvious wheel ruts where it had travelled. Vertical vibration level in field 4 was consistently above 20% lower when ploughing over the speed range from 1.8 to 2.8 m/s (4 to 6.3 mile/h). In field 3, measurements were made using a five furrow plough, and the results are compared with those for a three furrow plough in Fig. 9. At 2.2 m/s rrde vibration is about 30% lower with the five furrow plough than with the three furrow plough. This is because the vertical damping force on the plough is proportional to draught force, which is greater with the five furrow plough and also its line of action is further back and hence the moment acting on the tractor is greater. The measurements made with the fully mounted five furrow plough were compared with those made when a depth wheel was attached to the rear of the plough to assist depth control. Some of the plough weight was carried by the wheel and hence not all the vertical force on the plough
258
TRACTOR
RIDE
VIBRATION
acted on the tractor. This would be expected to reduce the damping effect of the plough, and the results in Fig. 9 show that it did. The ride level with the fully mounted plough was about 25 % lower than when the plough was fitted with a depth wheel at the rear. So far only vibration in the vertical direction has been discussed. In the longitudinal and lateral directions, no conclusive differences in ride level were found between the tractor alone and when ploughing. In fields 2 and 4 there was some evidence that the longitudinal vibration might be reduced slightly at the highest speeds (Fig. IO), but more results and preferably at even higher speeds would be necessary to confirm this. Longitudinal vibration may be expected to be reduced because the plough damps tractor pitch motion and, since the driver is not seated at the centre of pitch, his motion in the longitudinal direction should be affected. Some measurements were made with the tractor running on the land out of the furrow and without a plough. The ride levels in all three directions tended to be higher than when the tractor was in the furrow. Not enough results were taken to be conclusive but this trend might be expected since in the furrow two of the tractor wheels travel on a smoother surface than the land. Tractor
2 Forward
alone
3 speed
(m/s)
Fig. IO. Longitudinal acceleraion levels in fields 2 and 4
4.
Discussion
The results obtained in this work have several implications for future tractor design because both main conclusions become increasingly important at ploughing speeds above about 2.2 m/s (5 mile/h). Tractor ride vibration levels are significantly reduced when ploughing and because the damping forces are transmitted by the tractor linkage, draught controls which sense linkage force are less effective. Considering the amount of work that has been carried out into tractor ride dynamics, it is perhaps surprising that to date there has been no published work which includes the effect of implements or trailers. This work has shown that a plough significantly affects tractor behaviour and it is likely that other implements will also modify tractor behaviour though perhaps not to such an extent. Cultivators for example have similar soil force characteristics to ploughs except that the first term in the vertical force relationship which was assumed constant for the plough, is significant. Steady state vertical force increases with depth, so it opposes the damping term in the vertical force equation. In fact, if it is greater than the second term, negative damping would result and the system would be unstable. The damping term is proportional to ((draught force)/(forward speed) x rate of change of depth) so, as forward speed increases, the damping term decreases and stability decreases. The draught control response also affects stability, because when the linkage is moved, the vertical force changes, due to the rate of change of depth term. It is possible therefore for the draught control response to excite a natural frequency of the tractor and implement and increase tractor ride vibration.
D. A.
259
CROLLA
In general, any implement which is mounted on the tractor linkage will alter the tractor natural frequencies especially in pitch, because the implement weight acts well behind the tractor centre of gravity. Two-wheel trailers which transfer weight on to the tractor drawbar would also be expected to affect tractor ride significantly. There is scope for further work to investigate the effects of implements and trailers on tractor behaviour. The future of draught controls which sense linkage force must be questioned at high speed. Future trends in implement control will be influenced by the advances in cultivation implement design and in improvements to tractor ride at higher speeds. The benefits of improved ride level for high speed ploughing may be an important advantage o\er other cultivating techniques. The plough would have to be mounted and an improved draught control would be required since the performance of present linkage force sensing controls has been shown to be inadequate5 over about 2.2 m/s (5 mile/h). The force fluctuations due to implement damping are in the same frequency range as those due to implement depth variations, so they cannot be filtered out. A possible alternative is a control which senses draught force only, driveline torque, engine speed or wheelslip. A pure draught sensing control could be arranged without too much modification to present designs, by having two sensing units, one sensing top link force and the other sensing lower link force. The addition of the two signals, eliminates the vertical force and the resulting sum is equal to the draught force only. 5. Conclusions A theoretical model showed that when ploughing, the plough exerted considerable damping on the tractor pitch. Fieldwork confirmed this and tractor vertical ride vibration level measured at the driver’s seat was reduced by up to 50% when ploughing heavy land and 30% on light land. The significant effect of the plough on tractor behaviour indicates that work on tractor ride dynamics should be extended to include implements and trailers. Also the damping forces transmitted from the plough to the tractor cause fluctuation% in linkage force, and draught controls which sense this force receive spurious signals making them inadequate at speeds above about 2.2 m/s (5 mile/h). REFERENCES
Dwyer, M. J. The braking performance of tractor-trailer combinations. J. agric. Engng Res., 1970 15 149 Stayner, R. M.; Bean, A. G. M. Tractor Ride Investigations; a survey of vibrations experienced by drivers in fieldwork.
N.I.A.E. Departmental Note DN/E/578/1445 (unpublished), 1975
Dwyer, M. J. The dynamic performance of tractor-implement combinations. Proc. Instn Mech. Engrs, 1969-70 184 Part 3Q Crolla, D. A. A computer programme for simulating the field performance of a tractor and mounted cultivation implements under varying load condition. N.I.A.E. Note 62/1410 (unpublished), 1975 Crolla, D. A.; Pearson, G. Response of tractor draught controls to nmdom inputs. J. agric. Engng Res., 1975 20 181.
Hilton, D. J. A frequency weighted ride-meter. J. agric. Engng Res., 1970 15 379 Appendix A Values of parameters
The figures used are from a ballasted 63 p.t.o. kW tractor and three furrow plough. Tractor mass M 4300 kg J Tractor moment of inertia 2940 kg m2 Plough mass 545 kg Plough moment of inertia 727 kg m2 5 Front tyre stiffness 300 000 N/m k, Front tyre damping 725 N s/m Cf Rear tyre stiffness 400 000 N/m k,
260
TRACTOR
Rear tyre damping Hydraulic stiffness Hydraulic damping Top link sensing unit stiffness Top link sensing unit damping Distances (see Fig. 1)
Equivalent soil stiffness Equivalent soil damping
RIDE
3500 N s/m 1400 000 N/m 14 000 N s/m 2 500 000 N/m 0 lm lm O-5 m 0.79 m 1.43 m 0.27 m O-76 m 0.23 m 43 700 N/m 8700 N s/m
Appendix B The four second-order equations of motion can be written a,i’+a,~+a,B,+a,B,+a,i+a,~+a,B,+a,B,+a,z+a,,~=O blP+b,~+b,B;~+b,~,+b,i+b,~+b,8~+b~~~+bs~+b~o~+b~~~~+b~,~z=O c~i-+c,~+c,B,+c,ii,+c,i+c,~+C,B~+C*~2+CgZi-C~0~2=0 d,i+d,d;+d,B,+d,B,+d,i+d,~+d,81+ds82+d,z+d,,~+d,,B,+d,,8,=0
where
a,=M+m -ml, as=-
-ml, Cl=-
-ml1 2 -ml, a4=2 a5=cf+cr+c, a, = bc,--ac,-c,14 a, =-c,ll a8=-cslz a,=kf+k, aIo= bk,--ck, -ml, b,=2
as=-
b,=J+Ie+m1,2 b,=ml,l, b,=ml,l, b,=bc,-ac,+csll b,=b2cf+a2c,+cs1,2 b, = c,l,l, b8=cs121, b,=bk.,-ak,+k,(d+h, b,,=b2kf+a2k,-k,14 b,,=-ksl, (2d+h,) b,,=-k,l, (Tdfh,)
2
c,=ml,l, c,=ml12 c,=ml,12 cg=-c 1 cs = c,r,;t Cll12 c,=+c,112 4 r,=c,l,12
2
cg=-
kA2 4
+-q
(2d+h,)
d,=ml,l, d,=ml,l, d,=lo2+m122 d,=--c,l, d,=c,l,l, d,=c,l,l, d8=c2132+cs122 ds=k,(d+h,) 4o=-k,(dr,+dl,+h,l,) d,,=-k,l,(d+h,) d12=k,l,“-k,1,(2d+h,)
VIBRATION
D. A.
CROLLA
261
Combining these four equations with the four first-order equations gives the matrix equation
CA’$@I 1 + [B’l(v) = 0 where
y=
-2
--
k 0:
i -s:_ A’ E-
B' =
a, a, a3 a4 0 0 0 0 b, b, b, 6, 0 0 0 0 c,o 0 0 0 ~~~d,OOOO 00001000 00000100 00000010 _00000001 a5 a6 a, a, a, a,,0 0 b, b, b, b, b, blob,, b,, 0 yyj72dd>d 00 _; o6 o7 o8 o9 olooll 012 o-1 0 0 0 o-1 0 -0 0 o-1
0 0 0
0 0 0
0 0 0
0 0 0