Review of Models for Predicting Performance of Narrow Tillage Tool R. D. Grisso, John V. Perumpral ASSOC. MEMBER ASAE ABSTRACT OUR narrow tillage tool models are reviewed. Assumptions, capabilities and limitations associated with each model are discussed. Tillage tool performance under two different soil conditions have been predicted using the four models. Simulated results are compared with the experimental results. F INTRODUCTION During the last two decades several mathematical models have been developed for predicting the performance characteristics of tillage tools in soils. These include two-dimensional models for wide tools and threedimensional models for narrow tillage tools. In general, the validity of the individual models have been established by comparing predictions with experimental results. Despite exhaustive studies with individual models, no attempts have been made to compare these models. A comparison including assumptions, capabilities, and limitations associated with each model would be extremely valuable for all interested in tillage mechanics. Therefore, the objective of this paper is to compare selected models to predict the behavior of narrow tillage tools. Four models—developed by (a) Hettiaratchi and Reece (1967), (b) Godwin and Spoor (1977), (c) McKyes and Ali (1977), and (d) Perumpral et al. (1983) were selected for this study. A brief discussion of each model and a comparison of observed and simulated results from the different models is included. The tool angles and depths are limited to 0 to 90 deg with the horizontal and 0 to critical depth (6 x tool width), respectively. SOIL-TOOL INTERACTION MODELS Hettiaratchi and Reece (1967) The Hettiaratchi-Reece model is based on the Passive Earth Pressure theory. The analysis of three-dimensional failure assumes that the failure configuration is composed of forward and sideways failure regimes. The former refers to the failure ahead of the soil-tool interface, and the latter involves the horizontal sideways movement of the soil away from the center of the interface. The total force on the tool due to the threedimensional failure is the vector sum of the forces due to forward failure (Pt), the sideways force (Ps), and the adhesion force on the interface. Article was submitted for publication in September, 1984; reviewed and approved for publication by the Power and Machinery Div. of ASAE in March, 1985. Presented as ASAE Paper No. 81-1535. The authors are: R. D. GRISSO, Graduate Research Associate, Agricultural Engineering Dept., Auburn University, AL; and JOHN V. PERUMPRAL, Professor, Agricultural Engineering Dept., Virginia Polytechnic Institute and State University, Blacksburg, VA. 1062 MEMBER ASAE The relationship developed for Pt utilizes the earlier analysis of plane soil wedges in two-dimensional soil failure by Hettiaratchi et al. (1966). Assuming that the forward failure regime in front of a loaded interface extends the full width and depth of the interface, the expression for the force due to forward failure is expressed as: Pf = 7 D 2 B N 7 + c D B N c + AdDBNa + qDBNq* [1] The N-factors in this equation can be obtained from graphs available in Hettiaratchi et al. (1966). The forward failure force component (Pt) makes an angle d, the soil-metal friction angle, with the normal to the interface. The sideways failure force Ps is composed of cohesive and gravitational forces. This force can be expressed as: Ps = 7(d + q/T)2 bNS7 + cdbNsc [2] where, d = effective depth of failure wedge b = effective width of failure wedge N sy andN sc = dimensionless factors due to gravitation and cohesion, respectively. The N-factors in equation [2] depend on the roughness of the interface. Separate relationships are given for perfectly smooth (6 = 0) and perfectly rough (6 = <)>) interfaces and are computed utilizing the charts in Hettiaratchi et al. (1966). The N-factors computed are, therefore, suitable for predicting sideways failure force on vertical tools. For inclined interfaces, multiplication of equation [2] by the inclination factor ( K J is recommended. The inclination factor is given by the expression: t a n " 1 (sin a cot i//) where, i// = (45 + 0/2) [3b] The generalized relationship for the sideways force due to a sideways failure is: Ps = [7 (d + q/7) 2 bNS7 + cdbNsc] Ka [4] By combining equations [1] and [4] and including the adhesion force on the interface, the draft and lift •Notations common to all four models are defined in Table 1. © 1985 American Society of Agricultural Engineers 0001-2351/85/2804-1062$02.00 TRANSACTIONS of the ASAE TABLE 1. DEFINITION OF NOTATIONS COMMON TO ALL FOUR MODELS. Notations assuming a constant radius (r) for the side crescents and extending it to an angle rj, where: Definitions and units T? = cos" 1 | Dcota/r J A^ c 0 8 7 B D a q r 0 Fx Fz P N Soil-metal adhesional factor, N/cm^ Cohesional factor, N/cm^ Internal soil friction angle Soil-metal friction angle Unit weight of soil, N/cm^ Tool width, cm Tool depth, cm Tool or rake angle from forward horizontal Surcharge pressure on the soil-free surface, N/cra' Rupture distance from tool to crescent, cm Rupture angle from direction of travel Horizontal or draft force, N Vertical or lift force, N Total tool force, N Dimensionless earth pressure coefficients [7] To obtain a relationship for the force due to the side crescents, a wedge-shaped elemental volume and forces acting on it were considered. An equation for the passive force due to the elemental volume was developed and integrated to obtain the relationship for the total force by one of the side crescents. The relationships for the draft and lift forces on the tool were developed by combining the relationships for side crescents and the center wedge to give: F x = [yD2Ny + cDN c + q D N J [B + rsinT?] sin(a+5) + A d BD[N a sin(o: + 6) + cos(a)] [8] (vertical) forces on a tillage tine can be expressed as: F x = Pfsin(a + 5) + Pssina + AdDBcota [5] F z = Pfcos(a + 5)4- Pscosa + AdDB [6] F z = - [ 7 D 2 N 7 + cDN c + q D N J [B + rsinT?] cos(a + 5) - A d BD[N a cos(a + 5) - sin(a)] Hettiaratchi and Reece (1967) conducted two separate experimental investigations to examine the validity of the model. Comparison of experimental and simulated results for a vertical tine at low depth-to-width ratios showed that the model over-predicted the tool forces. However, the general shape of the predicted curve was similar to that developed from the experimental results. In the case of inclined tools, the model had a tendency to under-predict the tool forces. Godwin and Spoor (1977) The Godwin-Spoor model was developed to predict forces on narrow tillage tines with a wide range of depthto-width ratios. Two separate models were developed for tools operating at depths less v or greater than critical depth. For this review we chose only the model for tool depths less than the critical depth. The failure wedge considered for the analysis is shown in Fig. 1. The total wedge was divided into a center wedge with width the same as the tool width and two side crescents. They considered the forces contributed by each section to obtain the total force on the tool. The relationship developed by Hettiaratchi et al. (1966) for wide tools (equation [1]) was utilized to obtain the tool force from the center position of the wedge. The equation for the force due to the side crescents was developed by [9] The use of equations [8] and [9] requires prior knowledge of the rupture distance (r). Godwin and Spoor (1977) developed a graph using the information from Payne (1956), Payne and Tanner (1959), and Hettiaratchi and Reece (1967) to describe the relationship between distance ratio (rupture distance/depth) and tool angle. Godwin and Spoor (1977) conducted low speed verification tests in a sandy loam soil. The results of the experimental tests were compared with the predicted total force. The predicted draft force compared reasonably well with the experimental results while poor comparisons were obtained for the predicted vertical force. Generally the vertical force was overpredicted. However, as the depth to width ratio increased, the agreement between the two improved. McKyes and Ali (1977) The three-dimensional model by McKyes and Ali (1977) is similar to the Godwin-Spoor model. One major difference is that the McKyes-Ali model does not require prior knowledge of the rupture distance (r) for computing the forces on the tool. The curved failure surface from the tip of the tool was assumed to be straight and makes an angle ft with the horizontal. Considering an impending soil failure condition, a relationship for the tool force was developed in terms of the failure angle (p) and pertinent soil and tool parameters. Through appropriate mathematical operation the wedge which created the minimum passive force was determined. The rupture distance was assumed to be: r = D | cotjS + cota | [10] where, p is the unknown failure angle. Fig. 1—Three-dimensional failure wedge in front of narrow tools for depths less than critical depth (Godwin and Spoor, 1977). Vol. 28(4):July-August, 1985 Knowing the rupture distance, equation [7] is used to determine the crescent angle. In order to develop a relationship for the total force on the tool, as in the models described earlier, McKyes and Ali (1977) also considered the forces contributed by the center wedge and side crescents. Thus, the total draft 1063 force on the tool was expressed as: F = F [11] + 2 F _ The final expression for equation [11] is similar to equation [1], and the N-factors for the draft-force were defined as follows: N,7 X NL (r/2D) I 1 + 2r sin 7?/3B i cot(a + 5) + cot(j3 + 0) ) 1 + cotj3 cot(j3 + 0) \ \ 1 + r sin T?/B( . . .[13] cot(a + 5) + cot(]3 + 0) (r/D)(l+ r sin 7?/B) NL [12] cot(a + 8) + cot(j3 + 0) [14] center wedge were replaced by two sets of forces acting on the sides of the center wedge. As in McKyes and Ali (1977), the slip surface was assumed to be straight. The center wedge and various forces considered for the analysis are shown in Fig. 2. The force vectors R, SF 2 , and CF 2 shown on faces " a b c " and "def' are the forces replacing the side crescents. The equation for earth pressure at rest was utilized to estimate these forces. The rupture plane "ebcf' was assumed to be straight, making an angle ft with the horizontal. Plane " a b e d " is the interface, and on this plane adhesional force and total tool forces exist. Assuming that the wedge shown in Fig. 2 is in equilibrium, the following equations for the draft and lift forces were developed by summing the forces in the horizontal and vertical directions: F z = Pcos(a + 5') = W w + 2SF2sinj3 + SF lS in|3 To determine the failure wedge, the angle p was determined on a trial basis by minimizing the NyX function of equation [12]. The ft angle obtained from this process was used for the remaining computations. McKyes and Ali (1977) compared the N-factors from equations [12] through [14] with those for wide tools by setting B = °°. The comparison showed very close agreement for smooth tools (6 = 0). However, for tools with a rough surface and for tool angles greater than 90 -<j>, the N-factors obtained from equations [12] through [14] were much higher. To compensate for this poor a g r e e m e n t , M c K y e s a n d Ali (1977) s u g g e s t an interpolation procedure which is discussed in detail in their paper. McKyes and Ali (1977) also compared their simulated results with other models and experiments. In general, the results from the different methods agreed well. Even though the McKyes-Ali model may not be as rigorous as many other models already developed, it appears to be straight forward and it could be easily used for threedimensional soil-cutting problems. Perumpral, Grisso and Desai (1983) The model by Perumpral et al. (1983) is similar to models developed by McKyes and Ali (1977) and Godwin and Spoor (1977). However, the side wedges flanking the + 2CF2sinj3 + CF^injS + ADFsina - Qcosj3 . . . [15] F x = Psin(a: + 5') = 2SF2cosj3 + SF IC OSJ(3 + 2CF2cosj3 + CF1cosj3 + Qsin|3 - ADFcosa. . . [16] The 6' in equations [15] and [16] represents the soilmetal friction angle. In the models discussed previously a constant friction angle 6 was used. In this model, the friction angle is expressed as a function of tool angle. The relationship for 6' is shown in Table 2. Combining equations [15] and [16], a relationship for P can be written as: P= sin(j3 + 8' + a + 0) - ADF cos(j3 + 0 + a) + 2 SF 2 cos 0 + W w sin(0 + 0) j + 2 CF 2 cos 0 + CF 1 cos(0) [17] In terms of soil and tool parameters: - [ A d BD(l + h/D)] cos(/3 + 0 + a)/sin a + yAx [2K Q z sin 0 + B sin(0 4- (l)] + c cos 0 [2 At + BD/sin |3] sin(j3 + 0 + a + 8') [18] TABLE 2. SOIL PARAMETERS USED IN THE COMPUTER SIMULATION. SoU Parameters 7 c 0 Fig. 2—The proposed failure wedge and the forces (Perumpral et al., 1983). 1064 6' Perumpral et al. (1983) McKyes a n d Ali (1977) 0.0206 N / c m 3 0.015 N / c m 3 0.29 N / c m 2 0.023 N / c m 2 31° 35° 24° for 0 < 61° 23.3° 5 0 . 5 - .450 for (3 > 61° TRANSACTIONS of the ASAE m m Hettiaratchi & Reece (1967) o — — a Godwin & Spoor (1977) <>_—o McKyes & Ali (1977) • • perumpral et al. (1983) • Observed Data (Perumpral et al. Equation [17] can be expressed in the form of the previous models as: 500 P = 7BD 2 N 7 + cBDN c + A d BDN a [19] 250 where: A i 50 BD 2 N = 7 J 2K n z sin 0 + B sin(0 + |8) ( ' ; ——^-*sin(j3 + a + 0 + 5') 25 [20] L J 30 40 50 Tool Angle N = ](h 2 ——+ - — - X <* BD sing I sin (0 + CK + 0 + 6') N -(1 + h/D) cos(g + 0 + a) cos c = a [21] , I1500 coefficient of earth pressure at rest, expressed as: KQ = 1 - sin0 70 80 90 100 (Degrees) Fig. 3—Draft force - tool angle relationship for 5 cm wide tool at a depth of 5 cm. sin(j3 + a + 0 + 5')sin(a) where, K 0 = 60 . Hettiaratchi & Reece (1967) Godwin & Spoor (1977) -' McKyes & Ali (1977) (1983) P e r u m p r a l e t al> Observed Data (Perumpral et al., 1983) • 1250 [23] 1000 750 h z = height of soil heave in front of the tool at failure = average depth at which the centroid of the failure wedge is located from the soil surface, 500 Tool Depth [241 expressed as: z - (1/3)(D + h) A, = area of each side, " a b c " and Fig. 4—Draft force - tool width relationship for a vertical tool at a depth of 5 cm. "def\ Hettiaratchi & Reece (1967) Godwin & Spoor (1977) McKyes & Ali (1977) Perumpral et al. (1983) Observed Data (Perumpral et al., 1983)^ expressed as: A x - 0.5D 2 (1 + h/D) [(1 + h/D)cota + cotjS] . . . [25] In equation [18] all parameters except failure plane angle (p) are known. According to Passive E a r t h Pressure theory, passive failure takes place when the resistance by the soil wedge is at a minimum. This theory implies t h a t t h e wedge which c r e a t e s m i n i m u m resistance must be identified to determine the rupture angle. Mathematically, this wedge can be identified by solving the equation: 5 Tool Width dP/dj3 = 0 [26] Since closed form differentiation of equation [26] is rather complex, a numerical procedure was developed to minimize the function and to determine the failure plane angle (/?). Perumpral et al. (1983) conducted verification tests under laboratory conditions using artifical soil. The simulation results compared well with experimental data. RESULTS AND DISCUSSION The purpose of this study was not to select a best model but to compare the basis for the models, the assumptions involved, and the capabilities of the model. All models discussed are based on the Passive Earth Pressure theory. A majority of the assumptions involved with the models are the same as those associated with the Vol. 28(4):July-August, 1985 (cm) 7 5 (cm) Fig. 5—Draft force - tool width relationship for a vertical tool at a depth of 10 cm. earth pressure theory. The models discussed neglect the inertial forces and are suitable only for predicting the forces on a narrow tine moving at extremely slow speeds. Most models assume a constant soil-tool friction angle (d). On the other hand, based on the observations made during the preliminary laboratory tests, Perumpral et al. (1983) assumed the soil-metal friction angle to be a function of the tool angle. All models had one term to account for the force due to adhesion at the interface, with the exception of the McKyes-Ali model. To compare the capabilities of the models, tool forces for different tool geometry were predicted and the results are compared with the experimental data. Tests results obtained by Perumpral et al. (1983) and McKyes and Ali (1977) under two different soil conditions were used. For 1065 Hettiaratchi & Reece (1967) Godwin & Spoor (1977) McKyes & Ali (1977) Perumpral et al. (1983) Observed Data (Perumpral et al. , 1983) H e t t i a r a t c h i & Reece (1967) Godwin & Spoor (1977) o— -—o McKyes & A l i (1977) Perumpral e t a l . (1983) Observed Data (McKyes & A l i , m a • • • a——a 60 51) - 40 - 30 ^ 20 ^ - 10 IrlT^— 1977) ^ ^ " ^ ^ ^ , — - - " . ' ^ - • ^ . ' - * —^-J^ ' i ' 60 Tool Angle 5 Tool Width Fig. 9—Draft force - tool angle relationship for a 5 cm wide tool at a depth of 5 cm. (cm) Fig. 6—Lift force - tool width relationship for a vertical tool at a depth of 10 cm. —• —° —o "• (Degrees) 7.5 Hettiaratchi & Reece (1967) Godwin & Spoor (1977) McKyes & All (1977) Perumpral et al. (1983) Observed Data (Perumpral et al., 1983) Hettiaratchi & Reece (1967) Godwin & Spoor (1977) McKyes & Ali (1977) Perumpral et al. (1983) Observed Data (McKyes & Ali, 1977) u 125 600 400 50 200 25 0 -200 40 50 Tool Angle 60 Tool Width 70 (cm) Fig. 10—Draft force - tool width relationshp for a vertical tool at a depth of 5 cm. (Degrees) Fig. 7—Lift force - tool angle relationship for a 10 cm wide tool at a depth of 10 cm. H e t t i a r a t c h i & Reece (1967) m Godwin & Spoor (1977) a a o——o McKyes & A l i (1977) Perumpral e t a l . (1983) Observed Data (Perumpral e t a l . , A D O , • • • A O - A A 1983) A D h ^ ^ ^ - ^ ^ ^ O ODD L o O O 8 ^ D • 1 o H e t t i a r a t c h i & Reece (1967) Godwin & Spo or (1977) McKyes & AJ i (1977) Perumpral et a l . (1983) '"A A ^ ^ \ h - o § ^ y ^ D 8 o B I Observed r~ 1 5 1 10 Tool Depth (cm) I 15 1 Rupture Distance 1 (cm) Fig. 11—A relationship between predicted and experimentally observed rupture distance. Fig. 8—Lift force - tool depth relationship for a vertical, 10 cm wide tool. each model, computer programs were developed to simulate the tool performance. Using parameters given in Table 2, tool combinations (five tool angles, three tool depths, and three tool widths) under two soil conditions were obtained from the computer simulation. Comparisons of predicted results with experimental data from Perumpral et al. (1983) are shown in Figs. 3 through 8. Similar comparisons utilizing the experimental data from McKyes and Ali (1977) are shown in Figs. 9 and 10. The graphs shown were selected to indicate the effect of different tool parameters on the draft and lift forces. Plots of draft force as a function of 1066 different tool parameters under both soil conditions (Figs. 3-5, 9 and 10) clearly indicate that the Hettiaratchi-Reece model consistently overpredicts the draft force. In general, the simulated results using other models agreed well with the experimental results. Similar plots for lift forces (Figs. 6 to 8) show that the McKyesAli model did not agree well with the experimental observations. Lift force obtained using the HettiaratchiReece model showed good correlation with experimental data in all but one case (Fig. 7). Fig. 11 shows a plot between predicted and experimentally-observed rupture distance. In general, reasonable agreement was obtained. TRANSACTIONS of the ASAE CONCLUSIONS The conclusions of this study are as follows. 1. All models discussed in this paper can be conveniently programmed to obtain the draft and lift fores on slow-moving narrow tools under different soil conditions. 2. All models were found to be useful for predicting the rupture distance. 3. All of the models except the Hettiaratchi-Reece model gave reasonable predictions. The draft force agreed well with experimental observations in three cases. The Hettiaratchi-Reece model had a tendency to over-predict the draft force. 4. All models except the McKyes-Ali model predicted the lift force on the tool reasonably well. Vol. 28(4):July-August, 1985 References 1. Godwin, R. J. and G. Spoor. 1977. Soil failure with narrow tines. J. Agric. Eng. Res. 22(4):213-228. 2. Hettiaratchi, D. R. P. and A. R. Reece. 1967. Symmetrical three-dimensional soil failure. J. Terramech. 4(3):45-67. 3. Hettiaratchi, D. R. P., B. D. Witney and A. R. Reece. 1966. The calculation of passive pressure in two dimensional soil failure. J. Agric. Eng. Res. 11(2):89-107. 4. McKyes, E. and O. S. Ali. 1977. The cutting of soil by narrow blades. J. Terramech. 14(2):43-58. 5. Payne, P. C. 1956. The relationship between the mechanical properties of soil and the performance of simple cultivation implements. J. Agric. Eng. Res. l(l):23-50. 6. Payne, P. C. and D. W. Tanner. 1959. The relationship between rake angle and the performance of simple cultivation implements. J. Agric. Eng. Res. 4(4):312-325. 7. Perumpral, J. V., R. D. Grisso, and C. S. Desai. 1983. A soiltool model based on limit equilibrium analysis. TRANSACTIONS of the ASAE 26(4):991-995. 1067