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A torque evaluation for a rotary subsoiler
Soil & Tillage Research 84 (2005) 175–183 www.elsevier.com/locate/still
A torque evaluation for a rotary subsoiler Maciej Miszczak Warsaw Agricultural University, Department of Fundamental Engineering, Nowoursynowska 166, 02-787 Warsaw, Poland Received 22 February 2002; received in revised form 21 October 2004; accepted 22 November 2004
Abstract There are many models describing the work of rotary tillers, including torque evaluation. However, the work of a rotary subsoiler is different from that of a rotary tiller, which cuts the soil into pieces and throws them behind the machine. The rotary subsoiler acts on the soil with pressure similar to passive narrow tines. However, the existing models of forces acting on passive implements refer only to their linear movement. In this paper, a hypothesis has been made that the methods used in predicting the forces on soil cutting narrow tines may be applied to a rotary subsoiler, taking into consideration the subsequent positions of the working element in soil and including some corrections connected to the rotary movement of such a tine. On this basis a new mathematical model for predicting torque for a rotary subsoiler in different soils has been developed. Because no theoretical models describing the work of rotary subsoilers can be found, the prediction of the torque requirement may be very important for designers, as well as for other experts applying such machines in the field operations. After development the model was verified in experiments performed in a soil bin, which proved the model to be reliable. # 2005 Elsevier B.V. All rights reserved. Keywords: Tillage; Rotary subsoil; Narrow tine; Soil wedge; Soil loosening
1. Introduction Application of rotary subsoilers for deep tillage is a new concept, not existing in common tillage practices. Substituting of passive tillage implements by the rotary machines can be particularly justified by a higher efficiency of power transferring through the tractor driving shaft rather than through the tractor wheels, which leads to considerable slip of the drive wheels.
E-mail address: [email protected].
There are numerous models describing the work of rotary tillers, including evaluation of the needed driving torque (e.g. Bernacki, 1981; Hendrick and Gill, 1971a, 1971b, 1971c, 1974, 1978; Jacuk, 1971; Kanariew, 1983). However, the operation of a rotary tiller, which cuts soil, and leaves the loosed soil clods behind the machine is different from the operation of rotary subsoilers. The narrow working elements (tines) of rotary subsoilers, acting deeply, exert pressure on soil with their surface, causing soil disturbance and loosening, similar to the passive narrow elements action. The models describing operation of narrow passive implements usually
0167-1987/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.still.2004.11.011
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M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
incorporate an assumption that the implement forward movement is only linear, with no rotation in the soil (Godwin and Spoor, 1977; Hettiaratchi et al., 1966; Hettiaratchi and Reece, 1967; McKyes, 1978; Swick and Perumpral, 1988a, 1988b). In this paper, an assumption has been made that during the work of a straight working element of a rotary subsoiler at each infinitesimal moment of time, the subsoiler action on soil reflects the action of the narrow implement, set in such position as the rotary element. According to a majority of the models describing operation of the passive implements, it has been assumed that the implement working surface at each position covers the distance from the existing depth of the tine end immersion depth up to the free surface of soil, while its disturbances occur in a three-dimensional space— with consideration to the critical depth. However, during the work of the rotary element – differently as with passive element – the critical depths and then the outlines of the side part of the disturbed soil lump will vary in particular implement positions. For that reason, in order to consider the effect of the disturbed lump side parts, it has been assumed that above the critical depth on side surfaces of soil prism placed in the central part – corresponding to the implement width – there occur forces resulting from the pressure of adjacent soil layers, angle of internal friction and soil cohesion (Perumpral et al., 1983). These forces are taken into account in searching for minimal values of resistance. For the zone of three-directional disturbance, the new model has been developed by modification of previous ones and by assuming that soil disturbance includes the occurrence of Rankin’s zone, while the cutting zone is assumed to be limited by the section of a straight line inclined at an angle resulting from minimization of implement resistance. It has also been assumed that the effect of inertia forces can be considered negligible because in practice there is a lack of movement of the disturbed lump by the narrow working element of the rotary subsoiler. This paper presents a mathematical model of the operation of a narrow working element in soil, enabling the calculation of the required driving torque under various soil conditions and variable design parameters. An empirical verification of the model is also presented.
2. Methodology 2.1. Mathematical model The layout of the disturbed soil lump conformed to assumptions is presented in Fig. 1. The driving torque Mn needed for the work of rotary subsoiler working element at each position is determined by equation: ðR dmax Þ þ ð2=3Þdk sin a ðR dmax Þ þ ðd þ dk Þ=2 ; þ Fpk sin a
M n ¼ Fp
(1)
where R is the radius of the working element (m), a is the current angle of element setting (8), d is the instant depth of work (m), dk is the critical depth (m), and dmax is the maximum working depth of rotary subsoiler at vertical setting of its working element (m), while the remaining designations are shown in Fig. 2a. The value of force exerted by working element on soil in the zone of three-directional disturbance, perpendicular to its surface, is obtained on the basis of the equilibrium equation set for this zone, in which a mental division into parts S1 and S2 has been introduced (Fig. 2b). This division was considered only as a mental one in order to simplify the mentioned equilibrium equations. Justification of such procedure was fully proved in further investigations with the use of a filming method, where the occurrence of soil failure according to an introduced plane of mental division of the soil lump was not found. Such procedure was also applied by Hettiaratchi and Reece (1989). Solving a set of equilibrium equations one can obtain the equation for F p: ½G2 sinðb þ ’Þ þ ðFc2 þ 2Fbt2 þ 2Fbc2 Þ cos ’ Fp ¼
þ FR0 cosðb þ ’Þ Fa cosða þ b þ ’Þ cos d ; sinða þ b þ d þ ’Þ (2)
where w is the angle of internal friction (8), b is the angle of inclination of bottom plane of disturbed soil lump in three-dimensional zone (8), d is the angle of external friction (8). The values of particular forces are equal to the weight of soil: G2 ¼ gbS2 ;
(3)
M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
177
Fig. 1. Full instant contour of deformed soil lump for the assumed model.
the force of soil cohesion on lower surface: c b ðx=2 dk ctg aÞ ; F c2 ¼ cos b
where k is the angle of inclination of bottom plane of Rankin’s zone (8). (4)
where c is the soil cohesion (kN/m2), b is the width of working element (m), x is the maximal range of soil failure (m), g is the weight of soil (kN/m3), forces on side surface: F bc2 ¼ cS2 ;
1 F bt2 ¼ ðgK o d k S2 tg ’Þ; 3
(5)
adhesion force: ad dk b ; (6) sin a where ad is the soil adhesion (kN/m2), force from the half of Rankin’s zone rejected mentally, determined on the basis of equilibrium equations for that zone (S1): Fa ¼
gS1 b sinð’ þ kÞþð2cS1 þ2ðgKo dr S1 tg ’=3 Þ FR ¼
þ cxb=2 cos kÞ cos ’ cosð’ þ kÞ
The side surfaces are determined from equations: 1 1 1 1 S1 ¼ ðxdr Þ; S2 ¼ ðxdk Þ þ ðxdr Þ ðdk dr ctg aÞ; 4 4 4 2 (8) where particular measurements result from geometrical dependence’s as shown in Fig. 2. The coefficient of vertical force transfer on the horizontal pressures in the soil was assumed according to the dependence given by (Perumpral et al., 1983): K o ¼ 1 sin ’:
The force in soil failure zone below the critical depth was determined according to a method proposed by Goodwin and Spoor (1977)
: (7)
(9)
F pk ¼
bcN c ðd dk Þ þ 0:5K o gbN q ðd2 dk2 Þ : sin a
(10)
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M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
Fig. 2. Working diagram of rotary element in the three-directional failure zone: (a) geometrical dependences and the position of impact forces for particular zones in relation to working element, (b) relation of forces acting on the central part of considered soil lump.
According to this method, the coefficients Nq and Nc be determined from equations: Nq ¼
ð1 þ sin ’Þ expð2ðp=2 þ ’Þ tg ’Þ ; 1 sin ’ sinð2k þ ’Þ
N c ¼ ctg ’ðN q 1Þ:
(11) (12)
In analysing the performance of the working element of rotary motion consideration should be given to the previously loosened soil. For such conditions, it was estimated, that the value of cohesion ‘‘c’’ was lower by 50%. Another characteristic feature of working elements which rotate as well as move forward is the possibility
M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
of covering zones of soil disturbed by the previous tine (Fig. 3). For this purpose a correction coefficient was introduced; considering the ratio of covered zones to the total disturbed cross-section: S Sk : (13) S The total cross-section ABCD of disturbed soil is determined by the equation:
Kp ¼
S¼
0:5½dr2 ðctg k
ctg bÞ þ dk2 ðctg a þ ctg bÞ:
(14)
The disturbance cross-section covering the zone PCDW (or P0 DW, depending on intersection point of cycloid), limited by cycloid loop, described by the tine working previously in soil was determined as: Sk ¼ Sk1 Sk2 þ Sk3 ;
(15)
where Sk1 ¼ PEFWðor P0 EFWÞ Z yp R ¼
179
by solution of the relevant pairs of equations in cases, if such intersection occurs. Using the calculated coefficient Kp, the soil resistance in the previously disturbed zone was calculated to be equal to about 80% of undisturbed soil resistance. This leads to the expression for the force value in that zone: F 0 p ¼ F p ð0:2K p þ 0:8Þ:
(17)
As a result, the model enabling the determination of the driving torque needed in every position of rotary working element was obtained, based on two variables: inclination angle of soil shearing plane b (Fig. 3) and critical depth dk. These values can be found by minimization of the function determining the driving torque value. The mean value of driving torque for the whole rotor of the rotary subsoiler amounts to: Z aok z M nap ¼ M n da: (18) 2p aop In this expression, ‘‘z’’ is the number of tines working in the same plane. The calculations according to the presented model were carried out for subsequent values of the tine position angle.
yw
y arcsinðy=rÞ cos arcsin þ dy; R l Sk2 ¼ PEFKðor P0 EFKÞ
2.2. Soil bin experiment
¼ ½yp ðR dmax Þxp ; (16)
Sk3 ¼ PCDK ½yp ðR dmax Þ þ dr 2 h ðan þ ao ÞR ðR xp l
¼
dmax Þ ctg ao þ x dr ctg k þ
i
dr2 ctg k; 2
or 1 ð½y ðR dmax Þ2 Þ ctg k; 2 p where l ¼ u=v, u is the peripheral speed of working element (m/s), v is the speed of travel (m/s). The values xp and yp (or x0 p and y0 p ) are coordinates of intersecting point of lines BC or CD, determining the bottom surfaces of disturbed soil lump with cycloid, described by point Np being the end of ‘‘previous’’ tine. These co-ordinates can be found
Sk3 ¼
A special research model of the rotary subsoiler was made with two rectilinear working elements spaced at 908 (Fig. 4). The length of each element from the centre of rotation amounted to 0.448 m, while their width to 0.02 m. Each element was of T-bar shape and was attached to the disc so, as to obtain a radial setting of its working surface. Four extensometers were glued on each element in a full bridge system. Two working elements were used on the research stand. It was sufficient for a proper evaluation of the rotary subsoiler, since the ‘‘second’’ subsequent element worked under the same conditions as the remaining elements. Therefore, the results of the ‘‘second’’ element were used for evaluation of all the working elements. The model was verified in the soil bin of dimensions 9 m 1 m 0.5 m. On a measuring carriage of the bin, a gearbox with one input shaft and two output shafts was mounted. The input shaft was driven by a slow rotating hydraulic motor, while on one (horizontal) output shaft the research model
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M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
Fig. 3. Overlap of present failure zone and that deformed by preceding element: (a) intersection occurs in the first failure zone, (b) intersection occurs in the Rankin’s zone.
(disc with two working elements) was installed. The second (vertical) output shaft was equipped with a friction roller for the rope, with the ends fixed rigidly to solid supports of the soil bin. As a result of the rope being wound up by a friction roller, rotating and moving synchronously with the investigated element, its rotary speed was related to its advancing speed. Under the assumed model conditions it corresponded to a kinematic index l = 2.114, being the ratio between a circumferential speed at the working element tips and the ground speed. Besides, the two additional rollers were used, making it possible to
obtain kinematic index values of l = 1.762 and l = 2.643. The soil bin was filled with a weak loamy soil of mechanical composition presented in Table 1. The state of soil was changed by its compacting and Table 1 Mechanical composition of soil Fraction content by mass (mm) 0.1–1.0
0.02–0.1
<0.02
Solid phase density (g/cm3)
81%
12%
7%
2.5
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and measured values of torque requirement in particular repetitions, for the different positions of the working element. Based on the Student’s t-test it was found in all cases that at a significance level a = 0.05 there was no ground for rejection of a null hypothesis for the mean value of differences between the measured and calculated results (Table 3). Besides, a conformity test was performed between particular measurement results and their theoretical values. The a and b coefficients of linear regression were determined in the form: M theoretical ¼ a þ bM experimental ;
(19)
Fig. 4. The working disc of model rotary subsoiler: 1, working element, 2, support of working element, 3, mounting disc, 4, extensometers, 5, spacer.
with a hypothesis that the condition H0: a = 0, b = 1 is fulfilled at conformity of the results. This hypothesis was tested with the transformed equation (Elandt, 1964) of the form:
watering, while its compaction was checked with a cone penetrometer with a tip of base diameter 20.27 mm and apex angle 308 (ASAE S313.2). The soil moisture content was determined by the oven drying method. The angle of internal friction of soil and its cohesion was determined by the method of direct shearing. The angle of external friction was determined by moving over the soil a plate made of the same material as the working element. The soil bulk density was determined by taking the soil samples of 100 cm3. All the measurements were repeated six to nine times. The four sets of quantities characterizing the soil were distinguished and designated as G1–G4. The results obtained are presented in Table 2 with the mean and standard deviation values for the whole set of measurements.
¯ theoretical þ aðb 1Þ þ ðb 1Þ2 na2 þ 2nM ¯ 2theoretical Þððn 2Þ=2Þ ðSSR =b2 þ nM Fc ¼ ; SSE (20) where SSR is the sum of squares for regression, SSE is the sum of squares for error, n is the number of measurement points. The obtained values are presented in Table 3. All the test values indicate close conformity of the theoretical model with a set of measurement results obtained from the empirical experiment.
3. Results and discussion In order to verify dependences presented in the theoretical model investigations were carried out under four soil conditions and at three values of the kinematic index l. The maximum depth of working element of 0.23 m was used during investigations.
2.3. Statistical methods In order to verify the model, a significance test was performed for the differences between the calculated Table 2 Specification of the soil used Designation of soil state
G1 G2 G3 G4
Weight of soil (kN/m3)
Angle of internal friction (8)
Angle of external friction (8)
Cohesion (kN/m2)
g¯
sg
’¯
sw
d¯
sd
c¯
sc
18.1 16.1 19.2 17.6
0.23 0.18 0.26 0.20
32.2 26.4 34.1 27.2
0.78 0.61 0.75 0.64
22.1 22.1 19.5 19.5
0.48 0.48 0.52 0.52
2.12 1.13 2.00 1.06
0.05 0.03 0.04 0.04
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M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
Table 3 Results of statistical tests on measured and calculated values of torque requirement Soil
Kinematics index (l ¼ u=v)
Test values Calculated
Measured
Student’s t-test
Fisher’s Fc-test
Student’s t-test
Fisher’s Fc-test
G1
2.64 2.11 1.76
0.190 1.430 1.015
3.088 1.400 2.875
1.994 1.997 1.994
3.130 3.138 3.130
G2
2.64 2.11
0.664 1.873
0.341 2.653
1.997 2.007
3.138 3.182
G3
2.64 2.11 1.76
0.033 2.231 1.821
2.546 2.505 1.618
1.993 1.993 2.021
3.126 3.126 3.233
G4
2.11 1.76
1.560 1.418
3.069 1.025
1.993 1.995
3.126 3.135
The rotational speed of working element amounted to 6 rpm. The torque values were calculated on the basis of theoretical dependences for constructional data
corresponding to the investigated rotary subsoiler model and to the same soil conditions. Changes in driving torque were estimated as a function of positions of the working element during work. An example of these results is presented in Fig. 5. The empirical curves are quite similar to theoretical one. In an initial part of the diagram within the zone of working element penetration, the driving torque increases quickly, then it slightly decreases near the maximal torque. The maximal torque depends on the soil parameters and occurs in soil G1 and G3 at a working element inclination angle of about 758, while in soils G2 and G4 at an angle of about 858 to level. The same variations were obtained for the theoretical curves. When the maximal torque is reached, it decreases very rapidly; in part of measurements one can find a characteristic collapse of the torque, observed also in theoretical courses. The maximal values of driving torque depend practically on the soil parameters only, and do not depend on the kinematic factor within the investigated range. In the four soils investigated there were distinct differences in maximal torque values between the heavy soils (G1 and G3) and the light soils (G2 and G4).
4. Conclusions Fig. 5. Example of variation of driving torque according to experimental measurements and theoretical predictions.
The presented mathematical model of a narrow rotary element’s impact on soil enables a prediction to
M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
be made of the required driving torque for rotary subsoilers under various soil conditions and at variable design parameters. Application of the presented model to calculations of driving torque and power requirement can be a basis for selection of basic design parameters of rotary subsoilers, as well as for determination of the most appropriate conditions of their use.
References Bernacki, H., 1981. Teoria i konstrukcja maszyn rolniczych, t. I, cz 1 i 2. PWRiL, Warszawa, Poland, p. 417. Elandt, R., 1964. Statystyka matematyczna. PWN, Warszawa, Poland, p. 595. Godwin, R.J., Spoor, G., 1977. Soil failure with narrow tines. J. Agric. Eng. Res. 22, 213–228. Hendrick, J.G., Gill, W.R., 1971a. Rotary-tiller design parameters. Part I. Direction of rotation. Trans. ASAE 14 (4), 669–683. Hendrick, J.G., Gill, W.R., 1971b. Rotary-tiller design parameters. Part II. Depth of tillage. Trans. ASAE 14 (4), 669–683. Hendrick, J.G., Gill, W.R., 1971c. Rotary-tiller design parameters. Part III. Ratio of peripheral and forward velocities. Trans. ASAE 14 (4), 669–683.
183
Hendrick, J.G., Gill, W.R., 1974. Rotary tiller design parameters. Part IV. Blade clearance angle. Trans. ASAE 17 (1), 4–7. Hendrick, J.G., Gill, W.R., 1978. Rotary tiller design parameters. Part V. Kinematics. Trans. ASAE 21 (4), 658–660. Hettiaratchi, D.R.P., Reece, A.R., 1967. Symetrical three-dimensional soil failure. J. Terramech. 4, 45–67. Hettiaratchi, D.R.P., Reece, A.R., 1989. A slip-line method for estimating passive earth pressure. J. Agric. Eng. Res. 42, 27–41. Hettiaratchi, D.R.P., Witney, B.D., Reece, A.R., 1966. The calculation of passive pressure in two-dimensional soil failure. J. Agric. Eng. Res. 11, 89–107. Jacuk, E.P., 1971. Rotacionnyje poczwoobrabatuwajuszczyje masziny, vol. 2Maszinostrojenije, Moscow, Russia, pp. 16–19. Kanariew, F.M., 1983. Rotacionnyje poczwoobrabatywajuszczyje masziny i orudia, vol. 1Maszinostrojenije, Moscow, Russia, pp. 32–41. McKyes, E., 1978. The calculation of draft forces and soil failure boundaries of narrow cutting blades. Trans. ASAE 1, 20– 24. Perumpral, J.V., Grisso, R.D., Dessai, C.S., 1983. A soil–tool model based on limit equilibrium analysis. Trans. ASAE 4, 991– 1067. Swick, W.C., Perumpral, J.V., 1988a. A model for prediction dynamic soil–tool interaction. J. Terramech. 25, 43–56. Swick, W.C., Perumpral, J.V., 1988b. A model for predicting soil– tool interaction. J. Agric. Eng. Res. 25, 384–397.
A torque evaluation for a rotary subsoiler Maciej Miszczak Warsaw Agricultural University, Department of Fundamental Engineering, Nowoursynowska 166, 02-787 Warsaw, Poland Received 22 February 2002; received in revised form 21 October 2004; accepted 22 November 2004
Abstract There are many models describing the work of rotary tillers, including torque evaluation. However, the work of a rotary subsoiler is different from that of a rotary tiller, which cuts the soil into pieces and throws them behind the machine. The rotary subsoiler acts on the soil with pressure similar to passive narrow tines. However, the existing models of forces acting on passive implements refer only to their linear movement. In this paper, a hypothesis has been made that the methods used in predicting the forces on soil cutting narrow tines may be applied to a rotary subsoiler, taking into consideration the subsequent positions of the working element in soil and including some corrections connected to the rotary movement of such a tine. On this basis a new mathematical model for predicting torque for a rotary subsoiler in different soils has been developed. Because no theoretical models describing the work of rotary subsoilers can be found, the prediction of the torque requirement may be very important for designers, as well as for other experts applying such machines in the field operations. After development the model was verified in experiments performed in a soil bin, which proved the model to be reliable. # 2005 Elsevier B.V. All rights reserved. Keywords: Tillage; Rotary subsoil; Narrow tine; Soil wedge; Soil loosening
1. Introduction Application of rotary subsoilers for deep tillage is a new concept, not existing in common tillage practices. Substituting of passive tillage implements by the rotary machines can be particularly justified by a higher efficiency of power transferring through the tractor driving shaft rather than through the tractor wheels, which leads to considerable slip of the drive wheels.
E-mail address: [email protected].
There are numerous models describing the work of rotary tillers, including evaluation of the needed driving torque (e.g. Bernacki, 1981; Hendrick and Gill, 1971a, 1971b, 1971c, 1974, 1978; Jacuk, 1971; Kanariew, 1983). However, the operation of a rotary tiller, which cuts soil, and leaves the loosed soil clods behind the machine is different from the operation of rotary subsoilers. The narrow working elements (tines) of rotary subsoilers, acting deeply, exert pressure on soil with their surface, causing soil disturbance and loosening, similar to the passive narrow elements action. The models describing operation of narrow passive implements usually
0167-1987/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.still.2004.11.011
176
M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
incorporate an assumption that the implement forward movement is only linear, with no rotation in the soil (Godwin and Spoor, 1977; Hettiaratchi et al., 1966; Hettiaratchi and Reece, 1967; McKyes, 1978; Swick and Perumpral, 1988a, 1988b). In this paper, an assumption has been made that during the work of a straight working element of a rotary subsoiler at each infinitesimal moment of time, the subsoiler action on soil reflects the action of the narrow implement, set in such position as the rotary element. According to a majority of the models describing operation of the passive implements, it has been assumed that the implement working surface at each position covers the distance from the existing depth of the tine end immersion depth up to the free surface of soil, while its disturbances occur in a three-dimensional space— with consideration to the critical depth. However, during the work of the rotary element – differently as with passive element – the critical depths and then the outlines of the side part of the disturbed soil lump will vary in particular implement positions. For that reason, in order to consider the effect of the disturbed lump side parts, it has been assumed that above the critical depth on side surfaces of soil prism placed in the central part – corresponding to the implement width – there occur forces resulting from the pressure of adjacent soil layers, angle of internal friction and soil cohesion (Perumpral et al., 1983). These forces are taken into account in searching for minimal values of resistance. For the zone of three-directional disturbance, the new model has been developed by modification of previous ones and by assuming that soil disturbance includes the occurrence of Rankin’s zone, while the cutting zone is assumed to be limited by the section of a straight line inclined at an angle resulting from minimization of implement resistance. It has also been assumed that the effect of inertia forces can be considered negligible because in practice there is a lack of movement of the disturbed lump by the narrow working element of the rotary subsoiler. This paper presents a mathematical model of the operation of a narrow working element in soil, enabling the calculation of the required driving torque under various soil conditions and variable design parameters. An empirical verification of the model is also presented.
2. Methodology 2.1. Mathematical model The layout of the disturbed soil lump conformed to assumptions is presented in Fig. 1. The driving torque Mn needed for the work of rotary subsoiler working element at each position is determined by equation: ðR dmax Þ þ ð2=3Þdk sin a ðR dmax Þ þ ðd þ dk Þ=2 ; þ Fpk sin a
M n ¼ Fp
(1)
where R is the radius of the working element (m), a is the current angle of element setting (8), d is the instant depth of work (m), dk is the critical depth (m), and dmax is the maximum working depth of rotary subsoiler at vertical setting of its working element (m), while the remaining designations are shown in Fig. 2a. The value of force exerted by working element on soil in the zone of three-directional disturbance, perpendicular to its surface, is obtained on the basis of the equilibrium equation set for this zone, in which a mental division into parts S1 and S2 has been introduced (Fig. 2b). This division was considered only as a mental one in order to simplify the mentioned equilibrium equations. Justification of such procedure was fully proved in further investigations with the use of a filming method, where the occurrence of soil failure according to an introduced plane of mental division of the soil lump was not found. Such procedure was also applied by Hettiaratchi and Reece (1989). Solving a set of equilibrium equations one can obtain the equation for F p: ½G2 sinðb þ ’Þ þ ðFc2 þ 2Fbt2 þ 2Fbc2 Þ cos ’ Fp ¼
þ FR0 cosðb þ ’Þ Fa cosða þ b þ ’Þ cos d ; sinða þ b þ d þ ’Þ (2)
where w is the angle of internal friction (8), b is the angle of inclination of bottom plane of disturbed soil lump in three-dimensional zone (8), d is the angle of external friction (8). The values of particular forces are equal to the weight of soil: G2 ¼ gbS2 ;
(3)
M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
177
Fig. 1. Full instant contour of deformed soil lump for the assumed model.
the force of soil cohesion on lower surface: c b ðx=2 dk ctg aÞ ; F c2 ¼ cos b
where k is the angle of inclination of bottom plane of Rankin’s zone (8). (4)
where c is the soil cohesion (kN/m2), b is the width of working element (m), x is the maximal range of soil failure (m), g is the weight of soil (kN/m3), forces on side surface: F bc2 ¼ cS2 ;
1 F bt2 ¼ ðgK o d k S2 tg ’Þ; 3
(5)
adhesion force: ad dk b ; (6) sin a where ad is the soil adhesion (kN/m2), force from the half of Rankin’s zone rejected mentally, determined on the basis of equilibrium equations for that zone (S1): Fa ¼
gS1 b sinð’ þ kÞþð2cS1 þ2ðgKo dr S1 tg ’=3 Þ FR ¼
þ cxb=2 cos kÞ cos ’ cosð’ þ kÞ
The side surfaces are determined from equations: 1 1 1 1 S1 ¼ ðxdr Þ; S2 ¼ ðxdk Þ þ ðxdr Þ ðdk dr ctg aÞ; 4 4 4 2 (8) where particular measurements result from geometrical dependence’s as shown in Fig. 2. The coefficient of vertical force transfer on the horizontal pressures in the soil was assumed according to the dependence given by (Perumpral et al., 1983): K o ¼ 1 sin ’:
The force in soil failure zone below the critical depth was determined according to a method proposed by Goodwin and Spoor (1977)
: (7)
(9)
F pk ¼
bcN c ðd dk Þ þ 0:5K o gbN q ðd2 dk2 Þ : sin a
(10)
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M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
Fig. 2. Working diagram of rotary element in the three-directional failure zone: (a) geometrical dependences and the position of impact forces for particular zones in relation to working element, (b) relation of forces acting on the central part of considered soil lump.
According to this method, the coefficients Nq and Nc be determined from equations: Nq ¼
ð1 þ sin ’Þ expð2ðp=2 þ ’Þ tg ’Þ ; 1 sin ’ sinð2k þ ’Þ
N c ¼ ctg ’ðN q 1Þ:
(11) (12)
In analysing the performance of the working element of rotary motion consideration should be given to the previously loosened soil. For such conditions, it was estimated, that the value of cohesion ‘‘c’’ was lower by 50%. Another characteristic feature of working elements which rotate as well as move forward is the possibility
M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
of covering zones of soil disturbed by the previous tine (Fig. 3). For this purpose a correction coefficient was introduced; considering the ratio of covered zones to the total disturbed cross-section: S Sk : (13) S The total cross-section ABCD of disturbed soil is determined by the equation:
Kp ¼
S¼
0:5½dr2 ðctg k
ctg bÞ þ dk2 ðctg a þ ctg bÞ:
(14)
The disturbance cross-section covering the zone PCDW (or P0 DW, depending on intersection point of cycloid), limited by cycloid loop, described by the tine working previously in soil was determined as: Sk ¼ Sk1 Sk2 þ Sk3 ;
(15)
where Sk1 ¼ PEFWðor P0 EFWÞ Z yp R ¼
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by solution of the relevant pairs of equations in cases, if such intersection occurs. Using the calculated coefficient Kp, the soil resistance in the previously disturbed zone was calculated to be equal to about 80% of undisturbed soil resistance. This leads to the expression for the force value in that zone: F 0 p ¼ F p ð0:2K p þ 0:8Þ:
(17)
As a result, the model enabling the determination of the driving torque needed in every position of rotary working element was obtained, based on two variables: inclination angle of soil shearing plane b (Fig. 3) and critical depth dk. These values can be found by minimization of the function determining the driving torque value. The mean value of driving torque for the whole rotor of the rotary subsoiler amounts to: Z aok z M nap ¼ M n da: (18) 2p aop In this expression, ‘‘z’’ is the number of tines working in the same plane. The calculations according to the presented model were carried out for subsequent values of the tine position angle.
yw
y arcsinðy=rÞ cos arcsin þ dy; R l Sk2 ¼ PEFKðor P0 EFKÞ
2.2. Soil bin experiment
¼ ½yp ðR dmax Þxp ; (16)
Sk3 ¼ PCDK ½yp ðR dmax Þ þ dr 2 h ðan þ ao ÞR ðR xp l
¼
dmax Þ ctg ao þ x dr ctg k þ
i
dr2 ctg k; 2
or 1 ð½y ðR dmax Þ2 Þ ctg k; 2 p where l ¼ u=v, u is the peripheral speed of working element (m/s), v is the speed of travel (m/s). The values xp and yp (or x0 p and y0 p ) are coordinates of intersecting point of lines BC or CD, determining the bottom surfaces of disturbed soil lump with cycloid, described by point Np being the end of ‘‘previous’’ tine. These co-ordinates can be found
Sk3 ¼
A special research model of the rotary subsoiler was made with two rectilinear working elements spaced at 908 (Fig. 4). The length of each element from the centre of rotation amounted to 0.448 m, while their width to 0.02 m. Each element was of T-bar shape and was attached to the disc so, as to obtain a radial setting of its working surface. Four extensometers were glued on each element in a full bridge system. Two working elements were used on the research stand. It was sufficient for a proper evaluation of the rotary subsoiler, since the ‘‘second’’ subsequent element worked under the same conditions as the remaining elements. Therefore, the results of the ‘‘second’’ element were used for evaluation of all the working elements. The model was verified in the soil bin of dimensions 9 m 1 m 0.5 m. On a measuring carriage of the bin, a gearbox with one input shaft and two output shafts was mounted. The input shaft was driven by a slow rotating hydraulic motor, while on one (horizontal) output shaft the research model
180
M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
Fig. 3. Overlap of present failure zone and that deformed by preceding element: (a) intersection occurs in the first failure zone, (b) intersection occurs in the Rankin’s zone.
(disc with two working elements) was installed. The second (vertical) output shaft was equipped with a friction roller for the rope, with the ends fixed rigidly to solid supports of the soil bin. As a result of the rope being wound up by a friction roller, rotating and moving synchronously with the investigated element, its rotary speed was related to its advancing speed. Under the assumed model conditions it corresponded to a kinematic index l = 2.114, being the ratio between a circumferential speed at the working element tips and the ground speed. Besides, the two additional rollers were used, making it possible to
obtain kinematic index values of l = 1.762 and l = 2.643. The soil bin was filled with a weak loamy soil of mechanical composition presented in Table 1. The state of soil was changed by its compacting and Table 1 Mechanical composition of soil Fraction content by mass (mm) 0.1–1.0
0.02–0.1
<0.02
Solid phase density (g/cm3)
81%
12%
7%
2.5
M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
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and measured values of torque requirement in particular repetitions, for the different positions of the working element. Based on the Student’s t-test it was found in all cases that at a significance level a = 0.05 there was no ground for rejection of a null hypothesis for the mean value of differences between the measured and calculated results (Table 3). Besides, a conformity test was performed between particular measurement results and their theoretical values. The a and b coefficients of linear regression were determined in the form: M theoretical ¼ a þ bM experimental ;
(19)
Fig. 4. The working disc of model rotary subsoiler: 1, working element, 2, support of working element, 3, mounting disc, 4, extensometers, 5, spacer.
with a hypothesis that the condition H0: a = 0, b = 1 is fulfilled at conformity of the results. This hypothesis was tested with the transformed equation (Elandt, 1964) of the form:
watering, while its compaction was checked with a cone penetrometer with a tip of base diameter 20.27 mm and apex angle 308 (ASAE S313.2). The soil moisture content was determined by the oven drying method. The angle of internal friction of soil and its cohesion was determined by the method of direct shearing. The angle of external friction was determined by moving over the soil a plate made of the same material as the working element. The soil bulk density was determined by taking the soil samples of 100 cm3. All the measurements were repeated six to nine times. The four sets of quantities characterizing the soil were distinguished and designated as G1–G4. The results obtained are presented in Table 2 with the mean and standard deviation values for the whole set of measurements.
¯ theoretical þ aðb 1Þ þ ðb 1Þ2 na2 þ 2nM ¯ 2theoretical Þððn 2Þ=2Þ ðSSR =b2 þ nM Fc ¼ ; SSE (20) where SSR is the sum of squares for regression, SSE is the sum of squares for error, n is the number of measurement points. The obtained values are presented in Table 3. All the test values indicate close conformity of the theoretical model with a set of measurement results obtained from the empirical experiment.
3. Results and discussion In order to verify dependences presented in the theoretical model investigations were carried out under four soil conditions and at three values of the kinematic index l. The maximum depth of working element of 0.23 m was used during investigations.
2.3. Statistical methods In order to verify the model, a significance test was performed for the differences between the calculated Table 2 Specification of the soil used Designation of soil state
G1 G2 G3 G4
Weight of soil (kN/m3)
Angle of internal friction (8)
Angle of external friction (8)
Cohesion (kN/m2)
g¯
sg
’¯
sw
d¯
sd
c¯
sc
18.1 16.1 19.2 17.6
0.23 0.18 0.26 0.20
32.2 26.4 34.1 27.2
0.78 0.61 0.75 0.64
22.1 22.1 19.5 19.5
0.48 0.48 0.52 0.52
2.12 1.13 2.00 1.06
0.05 0.03 0.04 0.04
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M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
Table 3 Results of statistical tests on measured and calculated values of torque requirement Soil
Kinematics index (l ¼ u=v)
Test values Calculated
Measured
Student’s t-test
Fisher’s Fc-test
Student’s t-test
Fisher’s Fc-test
G1
2.64 2.11 1.76
0.190 1.430 1.015
3.088 1.400 2.875
1.994 1.997 1.994
3.130 3.138 3.130
G2
2.64 2.11
0.664 1.873
0.341 2.653
1.997 2.007
3.138 3.182
G3
2.64 2.11 1.76
0.033 2.231 1.821
2.546 2.505 1.618
1.993 1.993 2.021
3.126 3.126 3.233
G4
2.11 1.76
1.560 1.418
3.069 1.025
1.993 1.995
3.126 3.135
The rotational speed of working element amounted to 6 rpm. The torque values were calculated on the basis of theoretical dependences for constructional data
corresponding to the investigated rotary subsoiler model and to the same soil conditions. Changes in driving torque were estimated as a function of positions of the working element during work. An example of these results is presented in Fig. 5. The empirical curves are quite similar to theoretical one. In an initial part of the diagram within the zone of working element penetration, the driving torque increases quickly, then it slightly decreases near the maximal torque. The maximal torque depends on the soil parameters and occurs in soil G1 and G3 at a working element inclination angle of about 758, while in soils G2 and G4 at an angle of about 858 to level. The same variations were obtained for the theoretical curves. When the maximal torque is reached, it decreases very rapidly; in part of measurements one can find a characteristic collapse of the torque, observed also in theoretical courses. The maximal values of driving torque depend practically on the soil parameters only, and do not depend on the kinematic factor within the investigated range. In the four soils investigated there were distinct differences in maximal torque values between the heavy soils (G1 and G3) and the light soils (G2 and G4).
4. Conclusions Fig. 5. Example of variation of driving torque according to experimental measurements and theoretical predictions.
The presented mathematical model of a narrow rotary element’s impact on soil enables a prediction to
M. Miszczak / Soil & Tillage Research 84 (2005) 175–183
be made of the required driving torque for rotary subsoilers under various soil conditions and at variable design parameters. Application of the presented model to calculations of driving torque and power requirement can be a basis for selection of basic design parameters of rotary subsoilers, as well as for determination of the most appropriate conditions of their use.
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Hendrick, J.G., Gill, W.R., 1974. Rotary tiller design parameters. Part IV. Blade clearance angle. Trans. ASAE 17 (1), 4–7. Hendrick, J.G., Gill, W.R., 1978. Rotary tiller design parameters. Part V. Kinematics. Trans. ASAE 21 (4), 658–660. Hettiaratchi, D.R.P., Reece, A.R., 1967. Symetrical three-dimensional soil failure. J. Terramech. 4, 45–67. Hettiaratchi, D.R.P., Reece, A.R., 1989. A slip-line method for estimating passive earth pressure. J. Agric. Eng. Res. 42, 27–41. Hettiaratchi, D.R.P., Witney, B.D., Reece, A.R., 1966. The calculation of passive pressure in two-dimensional soil failure. J. Agric. Eng. Res. 11, 89–107. Jacuk, E.P., 1971. Rotacionnyje poczwoobrabatuwajuszczyje masziny, vol. 2Maszinostrojenije, Moscow, Russia, pp. 16–19. Kanariew, F.M., 1983. Rotacionnyje poczwoobrabatywajuszczyje masziny i orudia, vol. 1Maszinostrojenije, Moscow, Russia, pp. 32–41. McKyes, E., 1978. The calculation of draft forces and soil failure boundaries of narrow cutting blades. Trans. ASAE 1, 20– 24. Perumpral, J.V., Grisso, R.D., Dessai, C.S., 1983. A soil–tool model based on limit equilibrium analysis. Trans. ASAE 4, 991– 1067. Swick, W.C., Perumpral, J.V., 1988a. A model for prediction dynamic soil–tool interaction. J. Terramech. 25, 43–56. Swick, W.C., Perumpral, J.V., 1988b. A model for predicting soil– tool interaction. J. Agric. Eng. Res. 25, 384–397.