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Automatic control of tractor mounted implements— an implement transfer function analyser
J. ugric. Engng Res. (1969) 14 (2) 117-125
Automatic
Control
of Tractor
An Implement
Mounted
Transfer Function
ImplementsAnalyser
P. A. COWELL* Tractor draught control systems may be regarded as automatic regulators employing negative feedback. The problem of draught control is defined in control theory terms, in which it is shown that the implement itself forms an element in the loop. The paper describes an apparatus for determining the transfer function of an implement, information which is necessary if quantitative predictions of the dynamic behaviour of a draught control system are to be made. The apparatus is shown to have a second useful function in providing a standard input disturbance for evaluating the performance of draught control systems in general. 1.
Introduction
The invention and commercial development of draught control by Ferguion whereby the working depth or draught of tractor-mounted implements may be automatically regulated represents probably the first large scale application of automatic control in agricultural engineering. During the twentieth century and particularly after the 1920’s there has been an enormous increase in knowledge of the principles of automatic control, stimulated largely by military requirements. Little attempt has been made hitherto to apply established control theory to the tractor draught control system. Where an attempt has been made’ it has not been backed by adequate experimental work. Consequently a quantitative understanding of the behaviour of tractor draught systems is lacking. Experimental work reported in 1946 by Hawkins* on the behaviour of a draught control system showed it to have inferior control of depth in response to small field disturbances, when compared with implements whose depth was controlled by a depth wheel running on the soil surface. More recently, work3 has been carried out on two tractors manufactured in 1965, in which draught control was by top link force sensing. The depth control performance was shown to be unsatisfactory particularly at speeds above about 4 miles/h. The main reason for errors in depth in the latter case was shown to be caused in large part by the kinematic limitations imposed by the geometry of the linkage in relation to the dimensions of the tractor and plough and the terrain profile. The current situation is the result of many years of careful development work by tractor manufacturers, and a great number of useful modifications have been incorporated yet good field behaviour appears to remain an elusive goal. It seems reasonable to suppose that satisfactory performance at the design stage may only be ensured when a quantitative theory of the dynamic behaviour of implement control systems is available. In this paper the control problem is defined in control theory terms, in which it is shown that the implement forms an element in the control loop. To obtain a quantitative prediction of the depth control behaviour of draught control systems the transfer function of the implement must be known. A piece of apparatus is described whereby this transfer function may be obtained experimentally. The apparatus has a second function in that it provides a convenient means whereby draught control systems may be tested and their performance evaluated. 2.
The control loop
Fig. 1 shows in principle the main features of the simplest type of draught control system. A force signal is sensed in the top link, or in the case of lower link draught control, in the lower * Department
of Agricultural
Engineering,
University
of Newcastle
on Tvnr
117
118
AUTOMATIC
CONTROL
OF TRACTOR
MOUNTED
IMPLEMENTS
Fig. I. Simple draught-control system
links of the hitch. It is said that this signal gives a measure of implement draught or depth and, if it is maintained at a constant level, so implement draught and consequently working depth will also remain constant. The top link force is measured by the displacement it produces in spring S. This displacement actuates a control valve B, which causes oil to be pumped into or released from the lift cylinder C. The degree of top link displacement required for the valve to be in the neutral position is adjustable by means of a hand lever, H, which controls the position of Z. In this way the required top link force and hence working depth may be set by the operator. Lower link control works in a similar way except that the lower links are attached to a frame which is restrained from longitudinal movement by a control spring. In control terms the functioning of the draught control system may be described in Fig. 2. The error signal (x,) actuates the control valve which causes the pump to deliver oil, Q, to, or release it from, the lift cylinder. This in turn raises or lowers the implement relative to the Surface
disturbances
Oil supply
Manual setting
-<--
t
w
xe
Control valve ond pump
0
i
xo
[
I I
Feedback linkage
Implement
!\ Soil force disturbances
Fig. 2. Block diagram of draught-control system
Working depth
Yj
P. A. C’OWELL
119
KEY TO SYMBOLS A
Z b
b,,
d
g KI Kg L
LI, Lr-
4 I2 M m
Q R
s t
-= effective area lift cylinder = amplitude of harmonic input from scotch yoke = vertical height between share point and the cross-shaft -= vertical height between the share point and the cross-shaft of the T.F.A. when harmonic drive is at central position = height of P above line AB -= acceleration of gravity -= transfer function of lift linkage ~= reciprocal of control spring stiffness = horizontal soil force acting on implement = combined force component acting along lower links -~=combined vertical force component at the ends of the two lower links := horizontal distance between point P and the rear axle of the tractor :1 wheelbase of tractor m= amplitude of harmonic input == mast height of implement ==oilflow -= resultant of soil and gravitational forces acting on implement z 1 Laplace variable -= time
V W
0
= vertical soil force acting on implement = weight of vertical motion carriage (V.M.C.) = circular frequency of harmonic input (rad/sec)
WI = weightofimplement x, = horizontal distance from V to cross-shaft = horizontal distance from imple&V ment c. of g. to cross-shaft vertical co-ordinate of soil surface YG = at a point G directly beneath the plough cross-shaft _ii = vertical acceleration of implement in space ye = vertical distance between L and plough cross-shaft = working depth of implement Yr yp = vertical co-ordinate of point P YC = vertical displacement of plough cross-shaft from P vertical co-ordinate of front wheel .vF = ground contact point 4’0 = equilibrium working depth vertical co-ordinate of rear wheel YR = ground contact point Y, = displacement of ram U = force in top link U, = force in top link when implement is in equilibrium at its set depth. a = inclination of top link to horizontal
tractor (yJ. The actual depth of the implement in the ground (y,) depends on the ground surface profile and the attitude of the tractor. The system is thus subject to surface disturbances as shown. The implement itself forms the next element in the loop. The output from the implement is some force parameter (to be discussed later), to which must be added such soil force disturbances as may be encountered due to changes in soil resistance, buried obstructions, etc. The implement “force” is sensed by a spring located in one of the links of the hitch, and the displacement of this spring, after multiplication by a mechanical linkage (x,), is compared with the hand lever setting (xi). If a difference exists a signal is passed to the controller to initiate a correction. In order to calculate the behaviour of any control system the relationship between the output and the input of each and every element (its transfer function) in the loop must be known in the dynamic state. Considering first the control valve, this is usually non-linear. Wingate Hill4 has studied the static characteristics of a Massey-Ferguson control valve, which may be idealized as in Fig. 3. in which it can be seen that the characteristic on the “drop” side is dependent on downstream
120
AUTOMATIC
CONTROL
OF TRACTOR
MOUNTED
TS
IMPLEMEN
Fig. 3. Idealized steady-state flow characteristics of MF35 control valve
pressure. Since this pressure is governed by the implement, interaction between the implement and the control valve takes place as indicated by the dotted line in Fig. 2. The next element is the lift cylinder. This may be easily represented mathematically for
dyr
Adt=Q.
Taking the Laplace transform
[for y =o,
jr=0
when t
As Y,(s) = Q(s)
;,s,=;
. . . (1)
which is the transfer function for the lift cylinder. The lift linkage is again straight forward if only small displacements are concerned. that Y,=&Y, and
It follows
. . . (2)
$(s)=K,.
A similar equation relates the input and output of the control spring and feedback linkage. ...
@=K,.
The implement itself presents the greatest difficulty. Fig. 4 shows the forces acting on an implement in the dynamic state. L, and L, represent the components of the combined force acting in the two lower links.
Urn cos u-j-WI
:.
i
1-f
. 1
ucos +.$L-+;)w+)v.
x,+Vx,-L
yL=O
. . . (4)
P.
121
A. COWELL
Fig. 4. External forces acting on a mounted implement
Also
CF,=O U cos a+L-L,
cos f3=0
Since the angle of the top link spring, S, to the horizontal is usually very small, the actual force sensed is U cos a as in Eqn 4. Similarly, with lower link sensing, only the horizontal component of the force (LI; cos p) is sensed. Hence Eqn 5 represents the force signal in the latter case. For top link sensing the signal F= Ucos a. If U, is the top link force when the plough is in equilibrium at its set depth yO, then the relationship required is u “,“““; i
u” (s). 0
This requires a knowledge of how L and V vary with working depth y1 in the dynamic state. Static results of L and V against y, are available for a wide variety of implements, but none is available for situations where yi is continuously changing. This is important from a control point of view for if there should be any phase delay in the development of L or V as the implement moves up and down in the soil or if any attenuation of the amplitude of L and V occurs, this could have a profound effect on the behaviour of the closed loop system. For this reason a rig was designed and constructed specifically to study the variation of L and V in the dynamic state. 3.
The transfer function analyser The analyser is shown in Fig. 5. Its purpose is to measure L and V as the implement moves forward through the soil while being subjected to a vertical displacement. This is achieved by mounting the soil engaging part of the implement (plough mouldboard, cultivator tine, etc.) directly on to a cruciform leg. The cruciform leg is suspended within a rectangular frame by by three strain gauge dynamometers, two horizontal and one vertical. The vertical dynamometer yields V directly; the difference between the two horizontal dynamometer recordings yields L. Lateral forces on the implement are resisted by small linear ball bearings held in position by elastic bands. The transfer function may be studied by subjecting the implement to a sinusoidal displacement in the vertical direction as it moves forward through the soil at constant forward speed and constant mean depth, L and V being recorded simultaneously. In order to impart the vertical harmonic motion the rigid rectangular frame is fitted with ball bearing rollers making it into a vertical motion carriage (V.M.C.). This V.M.C. now runs on a B
122
AUTOMATIC
CONTROL
Fig. 5. The transfer function
OF TRACTOR
MOUNTED
IMPLEMENTS
analyser
pair of hardened ground steel rails mounted vertically on a chassis, which itself can be coupled directly to a tractor three-point hitch. The V.M.C. may now be driven freely up and down these rails, the harmonic motion being generated by a scotch yoke mounted on the chassis and driven, through suitable reduction gearing, by a 3 hp electric motor. An important feature of the analyser is the spring loaded bellcrank supporting the V.M.C. The V.M.C. when equipped with dynamometers and a 10 lbf tine weighs approximately 125 lbf. The instantaneous power required to drive it rises rapidly with frequency (w) and amplitude (A) and is given bv I
Wk.0
--(l-Xsin 0X cos at i hp= 550 g and the maximum instantaneous power requirement in the cycle occurs when
and cos ot= +d(l
.. .
-sin2 ot).
With an amplitude of 2 in and a frequency of 3 c/s the maximum instantaneous power requirement is 1.2 hp. A heavier implement would require a proportionately higher instantaneous power. Fluctuating power demands of this order would cause undesirable fluctuations in angular frequency, w. To reduce this inertia effect to a minimum the V.M.C. was supported by means of a
123
P. A. COWELL
bellcrank and powerful compression spring. If the natural frequency of oscillation of this system were now made equal to the desired operating frequency, then the only power needed to sustain the motion would be that required to overcome vertical soil forces and friction. In practice, when a natural frequency of 1 c/s was chosen it was found that between 0 and 2 c/s the 3 hp motor could cope adequately. A number of detail features of the design are perhaps worth recording. The drive from the electric motor (3000 revjmin) is taken through a Vickers Detroit variable speed gear giving infinitesimally variable ratios down to almost 2.85: 1. The drive then passes through a 7$ : 1 Radicon reduction gear and directly to a crank shaft producing crank speeds from 0 to --:-140 revimin (w= 14.7 rad/sec). The throw of the crank may be screw adjusted to give amplitudes of up to 2 in. The crank spigot engages a rectangular brass sliding block in a conventional scotch yoke mechanism to produce the final sinusoidal motion. The rig was designed to cater for a maximum horizontal load on the implement of 1500 Ibf, a minimum side load of 500 lbf and a maximum vertical load of 1000 lbf. The V.M.C. is carried on deep groove ball-bearings fitted with steel tyres made from EN31 ball race steel, ground and crowned on the outside periphery to avoid any edge loading induced by slight misalignment in mounting. To use ball-bearings directly on the flat rail would be to invite premature failure. The depth of the implement in the soil is measured by means of a depth wheel running on the soil surface, ahead of the implement. The difference between the height of the soil surface and that of the implement is measured by a rotary potentiometer mounted on a frame contiguous with the V.M.C. (and hence the implement). A spring tensioned chain anchored at one end of the contiguous frame and at the other to the depth wheel passes over a sprocket on the potentiometer. Both force and depth recordings are usually made on a U.V. recorder. The depth wheel itself is carried on a vertical round bar passing through a pair of linear ballbearings mounted in a housing and sealed at each end to exclude dirt. 4. Experimental procedure The method of procedure is to mount the T.F.A. on the rear of a tractor as in Fig. 5 and support it on an axle and pair of wheels. (At Newcastle the rig has only been used in an indoor soil tank with prepared soil but there seems little reason why it should not be used in the field notwithstanding the usual difficulties encountered with outside work.) Having adjusted the axle height to give the required mean depth the implement is drawn forward at constant speed whilst being subjected to a harmonic input of the desired amplitude. The only trials conducted so far have been in a soil tank using a flat tine 8 in wide, raked forward at 60” to the vertical, at a mean depth of 8 in. The soil used was a fine moist sand; it was cultivated and compacted with a two level vibratory compactor prior to each run. The tests were only of a preliminary nature and over the frequency range studied (0 to 8 rad/sec) no appreciable phase change or amplitude attenuation in the development of L or V was observed. However it is to be expected, particularly with a mouldboard plough body under field conditions, that the amplitude of V would vary considerably with increased input frequency. 5.
Using the T.F.A. to evaluate the performance of a draught control system
In order to evaluate the performance
of a control system it is usual to study its transient and harmonic responses to suitable stimuli, e.g. unit step, ramp or impulse input for transient response; harmonic input for harmonic response. Since the implement and the soil together form an element in the control loop, a true picture of the overall system response may only be obtained by having the system actually at work in the soil. A draught control system is in fact a regulator whose function is to maintain working depth constant in spite of soil force or surface disturbances. It is therefore of interest to know the response of the system to disturbances from these sources.
124
AUTOMATICCONTROLOFTRACTORMOUNTEDlMPLEMENTS
The nature of the surface disturbance may be seen in Fig. 6 which shows a tractor and mounted plough. P is a point fixed relative to the tractor and is the point at which the plough cross-shaft lies when the plough is at the required working depth on level ground. P lies at height d above a line drawn through the ground contact points of the front and rear wheels. It is assumed that the share point lies directly beneath the cross-shaft; the small departure from this assumption in practical cases has only a small effect. Datum
Fig. 6. Tractor andplough in work
From Fig. 6: . . . (7)
y,+yi=b+yc+yp
.
also
03) . (9)
yR-YF -=
.
(10)
...
(11)
.. .
(12)
(Y,+d)-YF
12
l,-tl2 y&Y,. 2
Since b-d=y, yi=yo AA
+
yc+
[
:+1
equili- linkage brium displacedepth ment
f(x+r,)-;f(x+r,+r,)-f(x). 1 Y
I
surface disturbance
Eqn 12 indicates the nature of the surface disturbance showing how it depends on the soil surface profile f(x), the dimensions of the linkage and the wheelbase of the tractor.
P.
125
A. COWELL
If it should now be possible to generate a standard “surface disturbance function” in Eqn 12, this would provide a useful means of comparing the response of draught control systems to surface disturbances in general. The T.F.A. can provide such a function as follows. The T.F.A. is mounted on the back of a tractor with 3 point linkage, and the supporting wheels and axle removed. A body or bodies of a mouldboard plough may now be attached to the tool holder of T.F.A. The implement is set to work on level ground. If the harmonic drive is now engaged the dimension b varies harmonically, b=b,+M
sin cot.
Since the tractor is on level ground f(x)=f(x+~~)=f(x+~~+~~) and Eqn 12 becomes _Yi=y,+y,+M
sin CM.
. . . (13)
The T.F.A., therefore, simulates a purely harmonic surface disturbance. In practice the control system is being asked to respond to the disturbance by raising and lowering the implement chassis to keep the working tool at constant depth. By measuring the amplitude ratio of and phase shift between the input displacement and the depth, an estimate of system performance may be obtained. If so desired other types of surface disturbance may be generated by imparting the required motion to the V.M.C. The problem with field testing of draught control systems is that of finding a field undulation of repeatable proportions and soil properties. It is much easier to find or produce a level piece of ground and to use the T.F.A. for simulating the surface disturbance. In this way it may be used for comparing the performance of draught control systems or as a development tool to assess the effect of design modifications. Acknowledgement
The author wishes to thank the Massey-Ferguson in the construction of the equipment described.
Manufacturing Company for their assistance
REFERENCES
Kawamura, N. Dynamic analysis of hydraulically controlled three point linkage. Translation of paper in J. Sot. agric. Mach., Japan, 1956, 18 (3) ’ Hawkins, J. C.; Boa, W. Ploughs and surface irregularities. Agric. Engng Rec., Oxf., 1946, 1, 146 3 Cowell, P. A.: Len, S. C. Field performance of tractor draught control systems. J. agric. Engng Res., ’
1967, 12 (3) 215 4 Wingate Hill, R. A study of the characteristics of a tractor hydraulic pump and valve. M.Sc. thesis, University of Newcastle-upon-Tyne, 1964 (unpublished)
Automatic
Control
of Tractor
An Implement
Mounted
Transfer Function
ImplementsAnalyser
P. A. COWELL* Tractor draught control systems may be regarded as automatic regulators employing negative feedback. The problem of draught control is defined in control theory terms, in which it is shown that the implement itself forms an element in the loop. The paper describes an apparatus for determining the transfer function of an implement, information which is necessary if quantitative predictions of the dynamic behaviour of a draught control system are to be made. The apparatus is shown to have a second useful function in providing a standard input disturbance for evaluating the performance of draught control systems in general. 1.
Introduction
The invention and commercial development of draught control by Ferguion whereby the working depth or draught of tractor-mounted implements may be automatically regulated represents probably the first large scale application of automatic control in agricultural engineering. During the twentieth century and particularly after the 1920’s there has been an enormous increase in knowledge of the principles of automatic control, stimulated largely by military requirements. Little attempt has been made hitherto to apply established control theory to the tractor draught control system. Where an attempt has been made’ it has not been backed by adequate experimental work. Consequently a quantitative understanding of the behaviour of tractor draught systems is lacking. Experimental work reported in 1946 by Hawkins* on the behaviour of a draught control system showed it to have inferior control of depth in response to small field disturbances, when compared with implements whose depth was controlled by a depth wheel running on the soil surface. More recently, work3 has been carried out on two tractors manufactured in 1965, in which draught control was by top link force sensing. The depth control performance was shown to be unsatisfactory particularly at speeds above about 4 miles/h. The main reason for errors in depth in the latter case was shown to be caused in large part by the kinematic limitations imposed by the geometry of the linkage in relation to the dimensions of the tractor and plough and the terrain profile. The current situation is the result of many years of careful development work by tractor manufacturers, and a great number of useful modifications have been incorporated yet good field behaviour appears to remain an elusive goal. It seems reasonable to suppose that satisfactory performance at the design stage may only be ensured when a quantitative theory of the dynamic behaviour of implement control systems is available. In this paper the control problem is defined in control theory terms, in which it is shown that the implement forms an element in the control loop. To obtain a quantitative prediction of the depth control behaviour of draught control systems the transfer function of the implement must be known. A piece of apparatus is described whereby this transfer function may be obtained experimentally. The apparatus has a second function in that it provides a convenient means whereby draught control systems may be tested and their performance evaluated. 2.
The control loop
Fig. 1 shows in principle the main features of the simplest type of draught control system. A force signal is sensed in the top link, or in the case of lower link draught control, in the lower * Department
of Agricultural
Engineering,
University
of Newcastle
on Tvnr
117
118
AUTOMATIC
CONTROL
OF TRACTOR
MOUNTED
IMPLEMENTS
Fig. I. Simple draught-control system
links of the hitch. It is said that this signal gives a measure of implement draught or depth and, if it is maintained at a constant level, so implement draught and consequently working depth will also remain constant. The top link force is measured by the displacement it produces in spring S. This displacement actuates a control valve B, which causes oil to be pumped into or released from the lift cylinder C. The degree of top link displacement required for the valve to be in the neutral position is adjustable by means of a hand lever, H, which controls the position of Z. In this way the required top link force and hence working depth may be set by the operator. Lower link control works in a similar way except that the lower links are attached to a frame which is restrained from longitudinal movement by a control spring. In control terms the functioning of the draught control system may be described in Fig. 2. The error signal (x,) actuates the control valve which causes the pump to deliver oil, Q, to, or release it from, the lift cylinder. This in turn raises or lowers the implement relative to the Surface
disturbances
Oil supply
Manual setting
-<--
t
w
xe
Control valve ond pump
0
i
xo
[
I I
Feedback linkage
Implement
!\ Soil force disturbances
Fig. 2. Block diagram of draught-control system
Working depth
Yj
P. A. C’OWELL
119
KEY TO SYMBOLS A
Z b
b,,
d
g KI Kg L
LI, Lr-
4 I2 M m
Q R
s t
-= effective area lift cylinder = amplitude of harmonic input from scotch yoke = vertical height between share point and the cross-shaft -= vertical height between the share point and the cross-shaft of the T.F.A. when harmonic drive is at central position = height of P above line AB -= acceleration of gravity -= transfer function of lift linkage ~= reciprocal of control spring stiffness = horizontal soil force acting on implement = combined force component acting along lower links -~=combined vertical force component at the ends of the two lower links := horizontal distance between point P and the rear axle of the tractor :1 wheelbase of tractor m= amplitude of harmonic input == mast height of implement ==oilflow -= resultant of soil and gravitational forces acting on implement z 1 Laplace variable -= time
V W
0
= vertical soil force acting on implement = weight of vertical motion carriage (V.M.C.) = circular frequency of harmonic input (rad/sec)
WI = weightofimplement x, = horizontal distance from V to cross-shaft = horizontal distance from imple&V ment c. of g. to cross-shaft vertical co-ordinate of soil surface YG = at a point G directly beneath the plough cross-shaft _ii = vertical acceleration of implement in space ye = vertical distance between L and plough cross-shaft = working depth of implement Yr yp = vertical co-ordinate of point P YC = vertical displacement of plough cross-shaft from P vertical co-ordinate of front wheel .vF = ground contact point 4’0 = equilibrium working depth vertical co-ordinate of rear wheel YR = ground contact point Y, = displacement of ram U = force in top link U, = force in top link when implement is in equilibrium at its set depth. a = inclination of top link to horizontal
tractor (yJ. The actual depth of the implement in the ground (y,) depends on the ground surface profile and the attitude of the tractor. The system is thus subject to surface disturbances as shown. The implement itself forms the next element in the loop. The output from the implement is some force parameter (to be discussed later), to which must be added such soil force disturbances as may be encountered due to changes in soil resistance, buried obstructions, etc. The implement “force” is sensed by a spring located in one of the links of the hitch, and the displacement of this spring, after multiplication by a mechanical linkage (x,), is compared with the hand lever setting (xi). If a difference exists a signal is passed to the controller to initiate a correction. In order to calculate the behaviour of any control system the relationship between the output and the input of each and every element (its transfer function) in the loop must be known in the dynamic state. Considering first the control valve, this is usually non-linear. Wingate Hill4 has studied the static characteristics of a Massey-Ferguson control valve, which may be idealized as in Fig. 3. in which it can be seen that the characteristic on the “drop” side is dependent on downstream
120
AUTOMATIC
CONTROL
OF TRACTOR
MOUNTED
TS
IMPLEMEN
Fig. 3. Idealized steady-state flow characteristics of MF35 control valve
pressure. Since this pressure is governed by the implement, interaction between the implement and the control valve takes place as indicated by the dotted line in Fig. 2. The next element is the lift cylinder. This may be easily represented mathematically for
dyr
Adt=Q.
Taking the Laplace transform
[for y =o,
jr=0
when t
As Y,(s) = Q(s)
;,s,=;
. . . (1)
which is the transfer function for the lift cylinder. The lift linkage is again straight forward if only small displacements are concerned. that Y,=&Y, and
It follows
. . . (2)
$(s)=K,.
A similar equation relates the input and output of the control spring and feedback linkage. ...
@=K,.
The implement itself presents the greatest difficulty. Fig. 4 shows the forces acting on an implement in the dynamic state. L, and L, represent the components of the combined force acting in the two lower links.
Urn cos u-j-WI
:.
i
1-f
. 1
ucos +.$L-+;)w+)v.
x,+Vx,-L
yL=O
. . . (4)
P.
121
A. COWELL
Fig. 4. External forces acting on a mounted implement
Also
CF,=O U cos a+L-L,
cos f3=0
Since the angle of the top link spring, S, to the horizontal is usually very small, the actual force sensed is U cos a as in Eqn 4. Similarly, with lower link sensing, only the horizontal component of the force (LI; cos p) is sensed. Hence Eqn 5 represents the force signal in the latter case. For top link sensing the signal F= Ucos a. If U, is the top link force when the plough is in equilibrium at its set depth yO, then the relationship required is u “,“““; i
u” (s). 0
This requires a knowledge of how L and V vary with working depth y1 in the dynamic state. Static results of L and V against y, are available for a wide variety of implements, but none is available for situations where yi is continuously changing. This is important from a control point of view for if there should be any phase delay in the development of L or V as the implement moves up and down in the soil or if any attenuation of the amplitude of L and V occurs, this could have a profound effect on the behaviour of the closed loop system. For this reason a rig was designed and constructed specifically to study the variation of L and V in the dynamic state. 3.
The transfer function analyser The analyser is shown in Fig. 5. Its purpose is to measure L and V as the implement moves forward through the soil while being subjected to a vertical displacement. This is achieved by mounting the soil engaging part of the implement (plough mouldboard, cultivator tine, etc.) directly on to a cruciform leg. The cruciform leg is suspended within a rectangular frame by by three strain gauge dynamometers, two horizontal and one vertical. The vertical dynamometer yields V directly; the difference between the two horizontal dynamometer recordings yields L. Lateral forces on the implement are resisted by small linear ball bearings held in position by elastic bands. The transfer function may be studied by subjecting the implement to a sinusoidal displacement in the vertical direction as it moves forward through the soil at constant forward speed and constant mean depth, L and V being recorded simultaneously. In order to impart the vertical harmonic motion the rigid rectangular frame is fitted with ball bearing rollers making it into a vertical motion carriage (V.M.C.). This V.M.C. now runs on a B
122
AUTOMATIC
CONTROL
Fig. 5. The transfer function
OF TRACTOR
MOUNTED
IMPLEMENTS
analyser
pair of hardened ground steel rails mounted vertically on a chassis, which itself can be coupled directly to a tractor three-point hitch. The V.M.C. may now be driven freely up and down these rails, the harmonic motion being generated by a scotch yoke mounted on the chassis and driven, through suitable reduction gearing, by a 3 hp electric motor. An important feature of the analyser is the spring loaded bellcrank supporting the V.M.C. The V.M.C. when equipped with dynamometers and a 10 lbf tine weighs approximately 125 lbf. The instantaneous power required to drive it rises rapidly with frequency (w) and amplitude (A) and is given bv I
Wk.0
--(l-Xsin 0X cos at i hp= 550 g and the maximum instantaneous power requirement in the cycle occurs when
and cos ot= +d(l
.. .
-sin2 ot).
With an amplitude of 2 in and a frequency of 3 c/s the maximum instantaneous power requirement is 1.2 hp. A heavier implement would require a proportionately higher instantaneous power. Fluctuating power demands of this order would cause undesirable fluctuations in angular frequency, w. To reduce this inertia effect to a minimum the V.M.C. was supported by means of a
123
P. A. COWELL
bellcrank and powerful compression spring. If the natural frequency of oscillation of this system were now made equal to the desired operating frequency, then the only power needed to sustain the motion would be that required to overcome vertical soil forces and friction. In practice, when a natural frequency of 1 c/s was chosen it was found that between 0 and 2 c/s the 3 hp motor could cope adequately. A number of detail features of the design are perhaps worth recording. The drive from the electric motor (3000 revjmin) is taken through a Vickers Detroit variable speed gear giving infinitesimally variable ratios down to almost 2.85: 1. The drive then passes through a 7$ : 1 Radicon reduction gear and directly to a crank shaft producing crank speeds from 0 to --:-140 revimin (w= 14.7 rad/sec). The throw of the crank may be screw adjusted to give amplitudes of up to 2 in. The crank spigot engages a rectangular brass sliding block in a conventional scotch yoke mechanism to produce the final sinusoidal motion. The rig was designed to cater for a maximum horizontal load on the implement of 1500 Ibf, a minimum side load of 500 lbf and a maximum vertical load of 1000 lbf. The V.M.C. is carried on deep groove ball-bearings fitted with steel tyres made from EN31 ball race steel, ground and crowned on the outside periphery to avoid any edge loading induced by slight misalignment in mounting. To use ball-bearings directly on the flat rail would be to invite premature failure. The depth of the implement in the soil is measured by means of a depth wheel running on the soil surface, ahead of the implement. The difference between the height of the soil surface and that of the implement is measured by a rotary potentiometer mounted on a frame contiguous with the V.M.C. (and hence the implement). A spring tensioned chain anchored at one end of the contiguous frame and at the other to the depth wheel passes over a sprocket on the potentiometer. Both force and depth recordings are usually made on a U.V. recorder. The depth wheel itself is carried on a vertical round bar passing through a pair of linear ballbearings mounted in a housing and sealed at each end to exclude dirt. 4. Experimental procedure The method of procedure is to mount the T.F.A. on the rear of a tractor as in Fig. 5 and support it on an axle and pair of wheels. (At Newcastle the rig has only been used in an indoor soil tank with prepared soil but there seems little reason why it should not be used in the field notwithstanding the usual difficulties encountered with outside work.) Having adjusted the axle height to give the required mean depth the implement is drawn forward at constant speed whilst being subjected to a harmonic input of the desired amplitude. The only trials conducted so far have been in a soil tank using a flat tine 8 in wide, raked forward at 60” to the vertical, at a mean depth of 8 in. The soil used was a fine moist sand; it was cultivated and compacted with a two level vibratory compactor prior to each run. The tests were only of a preliminary nature and over the frequency range studied (0 to 8 rad/sec) no appreciable phase change or amplitude attenuation in the development of L or V was observed. However it is to be expected, particularly with a mouldboard plough body under field conditions, that the amplitude of V would vary considerably with increased input frequency. 5.
Using the T.F.A. to evaluate the performance of a draught control system
In order to evaluate the performance
of a control system it is usual to study its transient and harmonic responses to suitable stimuli, e.g. unit step, ramp or impulse input for transient response; harmonic input for harmonic response. Since the implement and the soil together form an element in the control loop, a true picture of the overall system response may only be obtained by having the system actually at work in the soil. A draught control system is in fact a regulator whose function is to maintain working depth constant in spite of soil force or surface disturbances. It is therefore of interest to know the response of the system to disturbances from these sources.
124
AUTOMATICCONTROLOFTRACTORMOUNTEDlMPLEMENTS
The nature of the surface disturbance may be seen in Fig. 6 which shows a tractor and mounted plough. P is a point fixed relative to the tractor and is the point at which the plough cross-shaft lies when the plough is at the required working depth on level ground. P lies at height d above a line drawn through the ground contact points of the front and rear wheels. It is assumed that the share point lies directly beneath the cross-shaft; the small departure from this assumption in practical cases has only a small effect. Datum
Fig. 6. Tractor andplough in work
From Fig. 6: . . . (7)
y,+yi=b+yc+yp
.
also
03) . (9)
yR-YF -=
.
(10)
...
(11)
.. .
(12)
(Y,+d)-YF
12
l,-tl2 y&Y,. 2
Since b-d=y, yi=yo AA
+
yc+
[
:+1
equili- linkage brium displacedepth ment
f(x+r,)-;f(x+r,+r,)-f(x). 1 Y
I
surface disturbance
Eqn 12 indicates the nature of the surface disturbance showing how it depends on the soil surface profile f(x), the dimensions of the linkage and the wheelbase of the tractor.
P.
125
A. COWELL
If it should now be possible to generate a standard “surface disturbance function” in Eqn 12, this would provide a useful means of comparing the response of draught control systems to surface disturbances in general. The T.F.A. can provide such a function as follows. The T.F.A. is mounted on the back of a tractor with 3 point linkage, and the supporting wheels and axle removed. A body or bodies of a mouldboard plough may now be attached to the tool holder of T.F.A. The implement is set to work on level ground. If the harmonic drive is now engaged the dimension b varies harmonically, b=b,+M
sin cot.
Since the tractor is on level ground f(x)=f(x+~~)=f(x+~~+~~) and Eqn 12 becomes _Yi=y,+y,+M
sin CM.
. . . (13)
The T.F.A., therefore, simulates a purely harmonic surface disturbance. In practice the control system is being asked to respond to the disturbance by raising and lowering the implement chassis to keep the working tool at constant depth. By measuring the amplitude ratio of and phase shift between the input displacement and the depth, an estimate of system performance may be obtained. If so desired other types of surface disturbance may be generated by imparting the required motion to the V.M.C. The problem with field testing of draught control systems is that of finding a field undulation of repeatable proportions and soil properties. It is much easier to find or produce a level piece of ground and to use the T.F.A. for simulating the surface disturbance. In this way it may be used for comparing the performance of draught control systems or as a development tool to assess the effect of design modifications. Acknowledgement
The author wishes to thank the Massey-Ferguson in the construction of the equipment described.
Manufacturing Company for their assistance
REFERENCES
Kawamura, N. Dynamic analysis of hydraulically controlled three point linkage. Translation of paper in J. Sot. agric. Mach., Japan, 1956, 18 (3) ’ Hawkins, J. C.; Boa, W. Ploughs and surface irregularities. Agric. Engng Rec., Oxf., 1946, 1, 146 3 Cowell, P. A.: Len, S. C. Field performance of tractor draught control systems. J. agric. Engng Res., ’
1967, 12 (3) 215 4 Wingate Hill, R. A study of the characteristics of a tractor hydraulic pump and valve. M.Sc. thesis, University of Newcastle-upon-Tyne, 1964 (unpublished)