US :opy 1 A STUDY OF THE PLOW BOTTOM AND ITS ACTION UPON THE FURROW SLICE A THESIS Presented to the Faculty of the Graduate School OF Cornell University for the degree of DOCTOR OF PHILOSOPHY E. A. WHITE Reprinted from Journal of Agricultural Research, Vol. XII, No. 4, January 28, 1918 A STUDY OF THE PLOW BOTTOM AND ITS ACTION UPON THE FURROW SLICE A THESIS Presented to the Fac(tlty of the Graduate School OF Cornell University for the degree of DOCTOR OF PHILOSOPHY BY E. a: WHITE Reprinted from Journal of Agricultural Research, Vol. XII, No. 4, January 28, 1918 ,• u {J. k-t w* al jQ^-ioli A STUDY OF THE PLOW BOTTOM AND ITS ACTION UPON THE FURROW SLICE' By E. A. White, - Assistant Professor of Farm Mechanics College of Agriculture of the University of Illinois INTRODUCTION The most ancient records show that from a very remote period man has used the plow, in one form or another, to assist him in stimulating the earth to bring forth a more bountiful harvest. As has been the case in many other Hnes of endeavor, theory has trailed far behind observation and experience in developing this implement. In fact, as far as can be ascertained, it was not until the last half of the eighteenth century that any serious attempt was made to develop a plow bottom from a theoretical standpoint, and even then the productions of Jeffer- son, Lambruschini, Small, Rham, and others can not be considered as thoroughly grounded upon well-developed theories; rather their works should be looked upon as hypotheses (fig. i). Experience in the field generally proved that the machines designed by these men were not all that could be desired — for example, it is reported ^ that when Lam- bruschini's helicoidal moldboard was taken into the field for trial the driver of the draft animals immediately observed that the force required to move this plow was too great for the results obtained. To be sure, geometrically exact moldboards furnished the basis in many instances for more perfect developments, but the results obtained by empirical plow designers who worked in the field were so far superior to the results obtained by the men who worked in the laboratory that the theorists were soon completely outstripped and even held up to ridicule by the men who developed their machines in the hard school of experience, until at the present time we find special types of plow bottoms designed » Approved for publication in the Journal of Agricultural Research by the Director, Cornell University Agricultural Experiment Station. 2 The experimental work for this paper was done under the direction of Prof. H. W. Riley, of the De- partment of Rural Engineering, Cornell University, and the mathematical developments were prepared under the supervision of Prof. F. R. Sharpe, of the Department of Mathematics. In addition to the above, gratelul acknowledgments are given to the following: To Profs. James McMahoi. and Virgil Snyder, of the Department of Mathematics, for their most timely and helpful suggestions; to Mr. J. E. Reyna, Instructor in Drawing, College of Agriculture, Cornell University; and to Mr. L. S. Baldwin, Instructor in General Engineering Drawing, University of Illinois, for making the drawings. 'Lambruschini, R. d'un nuovo orecchio oa coltri. In Gior. Agr. Toscano, v. 6. p. 37-80. 1832. Journal of Agricultural Research. Vol. XII, No. 4 Washington, D. C. Jan- 28, 191S It Key No. N. Y. (Cornell)— 3 (149) I50 Journal of Agricultural Research Vol. XII, No. 4 to meet certain field conditions; but no well-developed theory is avail- able to serve as a guide in this work. This paper is an attempt to begin a fundamental analysis of the plow bottom and its work, in the hope that some light may be thrown upon the theory of this humble but perplexing machine, and other attempts stimu- lated to delve further into the secrets which are still to be revealed regarding the theory of this important implement. Empirical methods have given the world plow bottoms which work well. It is still to be hoped that scientific investigation can refine and further perfect, supple- ment as it were, the productions of experience. The work undertaken by the writer can be naturally divided into three parts: (i) A study of the forms of plow bottoms; (2) an attempt to Date Name Generatrix Directrixes Equation ofSurhct Small 5tn:iight line Straightline & Catenary Stephens Stnslghf line Straight line and arc of Circle ^^tanffiz); 1768 JefFerson Straight Une Straight lines l^//=^- teia Day/5 /Ire of Circle /Ires ofCin:le I63Z Lambruschini 5tra/gh/ Line Straight line and tieliK ^ = f-anfaz) 1639 W/theroiy andP/erce Arc ofCyc/o/c/ Arcs ofCycioiaf 1640 Rham Straight line Straight l/nes a' b^-^^^ 1340 Rham Straight- line Curves /852 Knox Straight line Arcs of Circles fFu/ed sunifce of el&hrh order /654 Gibbs Straight line Arcs of Circles f.^f.-/=o 1663 f\/1eaof Straightline Arcs of Circles a^^b'' C^ '^ /667 tiolbrook Straightline Srraight Ijhes a'^3' c'-' 1834 Jacobs /f port/or? ffTom each of £ S or/aces; each surface haWng 2 sets ofstralghf fine generators. Fig. I. — Diagram giving the generatrices, directrices, and equations of surfaces of historical plow bottoms . analyze the motion of the soil particles as they pass over the surface, and (3) a mathematical analysis of the surfaces of the most important historical plow bottoms which were designed to be geometrically exact. It was, and still is, hoped that a knowledge of just what the plow bottom is and how it performs its work will be of material assistance in developing a theory which will furnish a very definite basis for the proper design of this fundamental implement of tillage. FORMS OF THE PLOW BOTTOM A study of modern American-manufactured plow bottoms reveals the fact that a large number of these are so constructed that their surfaces contain sets of straight lines, each set consisting of an infinite number of straight lines, so related that an equation or equations satisfied by the coordinates of points on the surface can be found. jaa. »8, 1918 Study of Ploiv Bottoms I C I Plate 6, A, represents a bottom with two sets of straight lines. The few lines shown in the illustration indicate that through every point of the surface two straight lines can be drawn which lie wholly on the surface until they pass off the edges of the bottom. These straight lines furnish the basis for the proof that such a surface is a portion of an hyperboloid of one sheet (for the form of this surface see fig. 3 to 7) whose equation can be developed and studied with mathematical exactness. The method of developing this equation will be given later, but at present we are mainly interested in the fact that there is a classs of plow bottoms on whose surfaces lie sets of straight lines, and, further, that one equation can be developed which will approximately represent the working surface of such a bottom. Further study shows that the surfaces of other plow bottoms contain sets of straight lines, but that one equation will not completely describe such a surface. In Plate 6, B, a bottom is shown whose surface is com- posed of a portion of each of two surfaces. Plate 6, C, shows a similar bottom, but in this case the two surfaces merge into each other farther back upon the moldboard. In Plate 6, D, a class of bottoms is represented whose entire surfaces do not contain an infinite set of straight lines. It is true that the share and back end of the moldboard exhibit the same characteristics that the first two classes have shown, but the lines do not continue to the fore part of the moldboard. Plate 7, A, shows a plow bottom with a convex surface which has two sets of straight lines. The American-manufactured plow bottoms studied can thus be divided into three general classes: (i) A portion of one quadric surface; (2) a portion of each of two quadric surfaces, and (3) nonquadric sur- faces. Nearly all forged bottoms belong to classes i and 2 with the majority falling into class 2, while most of the cast bottoms belong to class 3. It should be noted, however, that some recently designed cast bottoms depart from the general characteristics of class 3 and show clearly the two quadric surfaces of class 2. The lines running in the general direction, front to rear, marked "/," (PI. 6, A) will be called longitudinal lines, and those running in the general direction, top to bottom, marked "t" (PI. 6, A) will be called transverse lines. For the purpose of studying the forms of the various surfaces under consideration, a machine, illustrated in Plate 7, B, was designed and built for measuring the space coordinates of any desired point.* By means of slots and a system of pulleys attached to the drafting board the cross- bar can be kept horizontal and be moved both laterally and vertically, while the drafting board is attached to a frame which can be moved ' Similar machines are described in the following publications: Ck)Ui,D, J. S., et al. report on Triais , OF PLOWS. In Trans. N. Y. State Agr. Soc, v. 27, pt. i. 1867, p. 426. 1868. Giordano, Federigo. le ricerche sperimentai,i di meccanica agraria. p. ho. Milano, 1906. 152 Journal of Agricultural Research Vol. XII. No, 4 backward and forward upon guides so marked that the board in all posi- tions will be squarely across the guides. When a plow bottom is properly- placed upon the platform the x, y, and z coordinates of any point upon the surface can thus be recorded upon coor- dinate paper fastened " upon the drafting board. / / Q i^zVz^a X y DEVELOPMENT OF THE EQUATION j(,y,z .^xy2' From a mathematical standpoint the sur- face shown in Plate 6, A, presents the problem of finding the equation of a surface, given two sets of straight-line generators. This can be done if the equations of any three lines in the same set are known. Select three lines ah, cd, and ef (fig. 2). Let Xj, y^, z\, and Xn, ^2. ^2 be the coordinates of two points upon line ab; 3C3, y^, z^, and x^,y^, z^ of two points upon line cd; and x^, y^, z^, and x^, y^, z^ of two points upon line e/. The equations of the lines ab, cd, and ef, are Fig. 2. and x- -■^1 y- -.Vi z - -21 Xn- -^1 }'2 ' ->'i Zo- -21 X.- -X3 y- ->'3 z - "2^3 ^4- -^.3 y^- ->'3 ^■t~ -23 X- -^5 y- ->'5 z - -^6 •>'5 (I) (2) (3) From (2) the following equation for a plane perpendicular to the XY- plane and containing the line cd is obtained : n=(x - %) ()/, - y^) - (y - y^) {x, - x^) = O. (4) Similarly from (2) the equation of a plane perpendicular to the YZ- plane and containing the line cd is «5=0'->'3) (24-2.0- (2-23) iyi-yz) = o. (5) From (3), the equation of a plane perpendicular to the Xl^-plane and containing the line ef is 2*6= {x - ^5) O'o - yd -iy-yd{\-xd=o. (6) Similarly, from (3) the equation of a plane perpendicular to the KZ-plane and containing the line ef is ^7= (y - yd (26 - 25) - (2 - 25) (je - yd = o. (7) Jan. 28, i9i8 Study of Plow Bottoms 153 Consider u^ = Au^. (8) where A is a constant. This is the equation of a plane which contains the intersection of planes (4) and (5); hence it contains the line cd. Similarly Uq = Bu^ (9) where B is a constant, is the equation of a plane which contains the line ef. If A and B have such values that the point x', y' , z' is on (8), (9), and (i), the line of intersection of (8) and (9) meets (i) and is a generator (see fig. 2). Hence, {y'-yz){^-z^)-{^'-z,){y,-y,)' ^'°'' and ^ {x'- %^ { y^ -y^-iy'- ys) JH - ^5) . ,. iy'-y.W,-^.)-{z'-z,){y,-y,) ' ^"^ ^l^^=t^::y^=t^=K; (12) ^1-^1 y^-yx ^2-21 ' where A' is a constant. From equations (12) %' = K{%^-%^)^x^ (13) y'=K{y2-yi)+yi (14) z' = K(z2-z,)+Zi (15) From equations (10), (13), (14), and (15) ^^ i[K(x2-Xi)+Xi-X3\(y,-ys))-i[K{y.,-yj)+yi-y3](x,-X3)) _ . ^. {[K{y2-yi)+yi-y3]{Zi-Z3))-{[K{22-Zi)+Zi-z3]{yi-y3)) ' ^ ^ and from equation (8) ^^^ (x-X3)iy,-y3)-(y-y3)iXi-X3) , . From equations (11), (13), (14), and (15) {\K{ x.-x;)-\-x^-x^{yf,-y^y)-(\K(y^-y^)-^y^-y^{x^-x^) and from equation (9) ^_ (.r - xQlJ'e - Vs) - (y - >'5)(^6 - ^5) / N (j'->'5)(2b-25)-(s-25)()'e-y5) ^ ^-^ Eliminating A,B, and /v from (16), (17), (18), and (19), we have the equation of a surface through the lines ah, cd, and ef. The equations are left in this form because numerical substitutions are more easily made 154 Journal of Agricultural Research voi. xii. no. 4 at this point than would be the case if the indica<:ed operations were first performed with the symbols.^ The general form of the equation resulting from the previous operations is ax^ + hy^ + cz' + 2fyz + 2gxz + 2 hxy -\-2lx-\- 2my -\-2nz-\-d=0. (20) To reduce equation (20) to its simplest form the axes must be trans- lated and rotated. TRANSLATION OF AXES^ The origin of equation (20) is translated to the center by putting x=x'+Xa, y=y'+yo, z+z'+z^; (21) he values of Xg, Vq, and Zo being obtained from the following: aXo-^hyo + gZo+l = (22) hxo + hyo+fzo + m = (23) gXo+fyo+cza + n = 0. (24) These substitutions give, after dropping the accents from x', y', and 2', an equation of the following form : ax"^ + by^ + c^ + 2/;'2 + 2qxz + 2 hxy -\-G = 0\ (25) where G=-lx^^myo-VnZo + d. (25a) ROTATION OF axes' Equation (25) can be further reduced by a rotation of the axes. This is accomplished by means of a cubic equation /f.^-(a+6+c)A;2 + (a6+ac + tc-f-(r-/r)^-^ = 0; (26) where D- a h 9 h b / 9 f c (26a) I^et the roots of (26) be k^, k^, and k^. The desired equation, after trans- lating and rotating the axes is A k,x^ + k.y + ^'322 4- ,-, -r- = O; * (27) ' A numerical problem is developed by this method upon pages 156 to 160. * Snyder, Virgil, and Sisam, C. H. analytic geometry op space,, p. 77. New York, 1914. 'Idem, p. 79. *Idem, p. 86. Jan. 28, 1918 Stvdy of Plow Bottoms 155 Skeleton. Hyperboloid of One 5haet Fig. 3. Section z=o. Fig 3. Fig. 4. Section y = 0, Fig. 3. Fig. s. o y Section x^>=o. Fig 3. Fig. 6. Hyperboloid of One Sheet, showing Lines upon the Surface Fig. 7. 156 Journal of Agricultural Research voi. xii. no. 4 where A = DG. (27a) The direction cosines X, ju, v, of the angles which the new X-axis makes with the original axes are obtained from the following : (a—ki)\ + hfjL+gv = (28) h\-{-{h-k^)n+jv = (29) pX+//i + (c-fe,)i; = C> (30) ^\+lx'' + v''=l. (31) Similarly, the direction cosines of the angles which the V- and Z-axe make, after rotation, with the original axes are found by substituting feg and ^3, respectively, for k^ in equations (28), (29), (30), and (31). When equation (27) was developed from the surface of a plow bottom having two sets of straight-line generators, it had the following general form: X^ /,,2 2^ This is the equation of an hyperboloid of one sheet, a vase-shaped figure, the skeleton of a section of which is shown in figure 3. When z = equation (32) becomes -^ + p=i, and the cross section through the plane z = (fig. 4) is an ellipse. When y = 0, the equation becomes x^ z- -2 — 2~ i> ^"^^ the section through the plane y = (fig. 5) is a hyperbola. \~ z^ Similarly, when x = 0, f^. — -1= i (fig- 6). Figure 7 indicates the two sets of straight-line generators which lie on the surface of an hyperboloid of one sheet.^ APPLICATION OF THE DEVELOPMENT TO A PROBLEM In order to^develop the equation which will describe the surface of a plow bottom, it is necessary to obtain the data called for in equations (16), (17), (18), and (19). This application of the development will be carried through for the bottom represented in Plate 6, A, which bottom was placed upon the machine shown in Plate 7, B, so that the origin of 'The constants a, b, and c ol this equation do not necessarily have the same numerical values as in previous equations. ' The method for obtaining the equations of any line on the surface is given in Snyder, Virgil, and SiSAM, C. H. Op. cit., p. 93. Jan. 28, 1918 Study of Plow Bottoms 157 coordinates came at O, figure 8. The plane y = contains the points O, m, and n; and the plane x = contains the points O and m and is per- pendicular to the plane y = 0. The plane 2 = is perpendicular to both the planes y = and x-=0. The axes are considered to be positive in the directions indicated by the arrowheads (fig. 8). Three transverse lines, ab, cd, and ef (fig. 8), were selected and the following data obtained: Fig. 8. Table J.— Values {in inches) developed for the surface of the plow bottom shown in Plate 6, A Xi= 2.84 >'i= 5-7 2i=i6.o Xo= 7-42 >'2= 3-78 22=19.0 ^3= 4-42 y,= 8.74 23=20.0 ^4= 8.54 >'4= 6.43 24=23.0 ^5= 9-7 J'5= 10.88 25=26.0 X6=I2.58 J6= 7-65 2^ = 28.0 When the above values are substituted in the equations already developed, From (16) -2.287^ + 13-83 K-15.7 • ^ = (33) 158 Journal of Agricultural Research Vol. XII, No. 4 From (17) From (33) and (34) From (18) From (19) From (36) and {2>7) K- A = ■X— 1.783/ + 20 i.299:i/+2-3i-35 -i5. 73g+io.o4>/- 13.832+ 1 1.95 X- 1. 177>'- 2.282 + 51.45 -1.584/C + 6.335 B B- K-7.29 — X— .892>'+i9.4 . 6197+2- 32.74 * 7. 29a; + 2. 587-6.3352 + 65.85 .1;+. 0887— 1.542 + 32.46 (34) (35) (36) (37) (38) By eliminating K from equations (35) and (38) the following equation for the surface of the plow bottom is obtained: 3.gx-+y^- + T,.4Sz"-7.ssyz-7-^^xz + 6.79xy +87. irc+ 1 20.757- 75.052+ 227. 25= a (39) Table II is compiled for purposes of checking the values computed from equation (39) with those obtained by measuring. Table II. — Values (in inches) for the surface of the plow bottom shown in Plate 6, A, obtained by measurement X computed z y X from Differ equation (39) ence. 10 2 2.9 2. 27 63 15 6 1-53 1.56 - 03 15 4 3-58 3-77 19 15 2 6.9 6.32 ,=;8 20 10 3-72 3-8 08 20 8 4-73 4. 76 - 03 20 4 7.83 7-94 13 25 12 8. 22 8.12 I 25 9 9.07 9.2 - 13 25 6 10.43 10. 46 — 03 30 10 14 13.86 14 32 9 16.5 16. I 4 Jan. 28, 1918 Study of Plow Bottoms 159 To find the geometric center, substitute the coefficients from equation (39) into equations (22), (23), and (24). Solving, we find Xo= — 1.405 inches. j/o= 6.52 inches. 2^0= 16.4 inches. This translation of axes is shown in figure 9. From equation (25a) G=— 57.3. From (25) the equation of the surface referred to parallel axes through the center is 3.9a;2 + ;^'2 4- 3452^- 7-53>'2- 7-28x2+ 6.79x^-57.3 = 0. (40) k \y ^"^ ^ ^" equation (27), we find io.27x^+ .i28)r — 2.052:^ = 57.3 or 9 « *> The direction cosines of the angles which the axes make after rotation with the original axes are obtained by making the proper substitutions in equations (28), (29), (30), and (31). i6o Journal of Agricultural Research Vol. XII. No. 4 For the K-axis For the Z-axis For the X-axis 7= T0.6136 /!= T0.48 u= ±0.627. 7= ±0.7515 /x==Fo.i437 u= ±0.6445. 7= =Fo.i4i5 /x= ±0.828 y== ±0.5425. Figure 10 shows the axes after translation and rotation and the por- tion of the hyperboloid of one sheet which is a close approximation to the surface of this plow bottom. SURFACES ONE PORTION FROM EACH OF TWO QUADRIC SURFACES Fig. 10. were obtained from the share board. By the use of the method which has just been em- ployed to develop the equation of the surface of the plow bottom shown in Plate 6, A, two equations can be developed which will approx- imately represent the surface of the bottom shown in Plate 6, B. By taking the origin as at O, figure 8, the data of Tables III and IV and the front portion of the mold^ Table III- — Values {in inches) developed for the surface of the share and front portion of the moldboard of the plow bottom shown in Plate 6, B x,= 3.92 >■!= -8 2i= 8.0 .X2= 7-4 >'2= -75 2._>=I2.0 ^3= 1-73 >'3= 2.67 23=12.0 ^4= 6.78 3'4= 1-75 24=16.0 %= 2.36 25 = 15-0 ^6= 587 )'6= 2.7 26=17.0 0.25*' + 2. 34>^ +0.462'— 3.25^2— o.773(;2 + 2.66x>/ + 6.88% + 32.3>'- 5.812- 4.4 = (43) Jan. 28, 1918 Study of Plow Bottoms 161 Table IV. — Values {in inches)for the surface of the share and front part of the moldboard of the plow bottom shown in Plate 6, B, obtained by fneasuring 2 y X X computed from equa- tion (43) Difference 10 I 4-75 4-75 0. 00 10 2 I 1-75 8-37 I- 54 9. 00 . 21 - -63 15 2 5-47 5-64 - .17 ^5 3 3-77 3.82 - -05 '^S 4 I. I 1-3 . 2 From the remaining surface of the moldboard the following data of Tables V and VI were obtained : Table V. — Values {in inches) of rest of surface of moldboard shown in Plated, B x,= 8.67 X2= 4-96 •1^:3=1 1 -73 3'i= 4-95 .r,= 8.64 >3= 4.81 2, = 24.0 £2=22.0 23 = 29.0 x^= 9.08 .r5= 13.62 .r6=I2.24 ^4= 9° >'5= 6.23 .r6=ii.89 24 = 27.0 25=33-0 2 1 ^ >> \ \N \ \ \\ V A \ \\ \ \ \ \^ ^ N \ \ \ \ \ \ 1 \^ V \ ^ \ \ V \ \ \ \^ \ \ \ \ \ \ \ 1 \ \ \ \ \ ^ \ \ 1 \ \ \ \ \ \ \ \ 1 i \ \ \ \ \ \ \ \ \ \ \ \ i \ 1 1 , 1 1 1 i i J i ~! 1 40 (D) That the maxi- mum difference, z — s, ^3 for each path decreased uniformly across the 36 furrow slice. Thus, for Row I, re = 0.85 inch, the 34 maximum 2; — i- = i .05 inches, and when ic=i3.6 inches, the width of the furrow- slice, the maximum z—s=0\ so when x=7.5 inches, the max- imum z—s for Row V is 0.45 inch. (E) That the stretch- ing in each row took place uniformly up to the maximum point and then decreased uniformly until it was ^'^ zero when 2 = 5 = 40. (F) That the maxi- '^ mum stretching occurred midway be- 16 tween the point where the soil particle passed 14 upon the plow bottom and the point ^• = 40. 12 Thus, for Path I where the soil particle passed 10 upon the moldboard at the point 5- = 0.6: 32 30 28 26 24 22 8 Fig. 14. — Projection of the paths shown in Plate 9, A, upon the plane y—O. 40 — 0.6 = 39.4 inches. 39.4 -^ 2 = 19.7 inches. 19.7 + 0.6=20.3 inches. For Path I the point of maximum stretch- ing was at .y=2o.3 inches. The computations below show that for Path V, v/here the soil jaa 28.1918 Study of Plow Bottoms 167 particle passed upon the share at the point s=ii.6, the point of maximum stretching occurs at ^=25.8 inches. 40 — 11.6 = 28.4 inches. 28.4H-2 = 14.2 inches. 14.2+ 11.6=25.8 inches. The following is the simplest form of a function which meets the requirements imposed by the above conditions and, when the constants are determined, will describe the relations between z and i- for a soil particle on the bottom of the furrow slice as it passes over the surface of the plow bottom : z-s=^a{s^- + bs+cy (48) From equations (47) and (4S) z-vt = a[{vty + bvi+cy; (49) From (49) -^ and -~, the velocity and acceleration, respectively, of a soil particle in the z direction can be obtained. From equation (46) by differentiation we have {2ax-\-lz+m)£^+i2bz+lx+n)j^ = 0; (50) and .<^'X , / dx ,dz\dx (2ax+/.+ m)^^+(^2a^ + /^j^ ■d-z / .dz , Jx\dz ^ - . + i,bz+lx+n)^, + l^2b^^ + l^)^^=^0. (51) Similarly from equation (45) we find and ,^dx , , , ^dv ^ / V {2ax + l)-r-+(2by + m)-fj = 0; (52) (2a.. + /)g+ 2a(^;)+ i2by + m/^+ 2.(1)= O. (53) ^ , , . . dx dy , , From equations (50), (51), (52), and (53) the velocities j^, ^-, and the accelerations -rv, ,v of a soil particle on the bottom of the furrcw slice at at-' '■ can be obtained when -jj and j^ ^^^ known. In this problem, however, we are interested in the accelerations in the directions of the normal to the surface, designated by ".V," the tangent to the soil path "T," and the perpendicular to the plane formed by the normal and the tangent " R." 1 68 Journal of Agricultural Research voi. xii, no. 4 We can find Xj, ^Uj, v^, the direction cosines of the angles which A^ makes with the X-, Y-, and Z-axis in either of the following ways : If (20) (the equation of the surface of the plow bottom) is known, we have by differentiation \ ^1 ax^ + byo + gzQ + l hx^ + byg + fzo + m V, I (54) / l{aXf^ + hy^ + gzQ + iy+ {hx^ + by^ + fZf, + mf' or if the paths of the soil particles are known but the equation of the surface is unknown the angle Ny can be measured by means of a pro- tractor and plumb bob, as shown in Plate 9, C. The direction cosines Xi and Vy can then be computed from the following: (Xir+(Mi)^+K)^=i (55) dx dy ^ dz ^'dt^^Ht-^'^dr^'' ^56) where the values for tt, ,' , and ~ can be obtained from (49), (50), and (52). dx dy d'' The direction cosines of T (X,, 1J.2, u,) are proportional to -77, A, and ,"• Hence X2 At2 ^2 I dx dy dz (dx\.(dj\(dz\ (57) dt df dt y\dt)'^\dt )^\dt) The direction cosines of T (X3, /Xg, v^ can be computed from the follow- ing:* (X3)'+(M3)'+(^'3)*=I- (58) —^ = ^^ - ^^ =±i.' (59) ^3*^2— i'3M2 Mai'i-'^sA'i Mii^2-i^iM2 The components in the directions A'^, T, and R of the forces acting on a soil element of mass M, moving with the component accelerations d}x d^y . d^z dP'd^^'^'^df^^'^ d'X d^v d^z FN = M(X,-i^,+M,^' + i',^) (60) FT=M(4f + 4>'+„,|f) (6x) Fr=.M(X,^ + 4? + v,^^). (6.) ■Snydbr, Virgil, and Sisam, C. H. Op. cit., p. 40. Jan. 28, 191S Study of Plow Bottoms 169 EVALUATING THE CONSTANTS IN EQUATIONS (48), (46), AND (45) The methods of evaluating the constants in equations (48), (46), and (45) for a given soil path will now be considered. For this purpose Path V (PI. 9, A) will be taken. The general form of equation (48) is 2-.' = o. 2. This changes the form of the equation to a{xy + b{yy + l,x' + 7ny^0. (67) Taking a=i, three constants remain to be evaluated. From the trace of Path V upon the surface of the plow bottom, x' y' I 3-1 4 5-45 7 6.68 Substituting these values of %' and y' in equation (67) gives a= I 6= 4. 29 /,= -3o.85 mi= — 3. 67 (;c0== + 4.29(3^'r-3o.S5x'-3.67/ = O. (68) The axes are translated back to the original origin by substituting x^x'-j.Ci^ y = y'—o. 2 in equation (68), which gives ^2H-4.29>'--46. i5x-5.39>/+295.45 = 0. (69) Numerical Example The surface of a plow bottom is represented by the equation 0.54a;"— i.-,2y-+ i.i2z- — 3.6gyz— i.62xz + 2.o4xy + 53.6^x+ ii^.goy — 46.4Z+ J^9.4. = 0. The motion of a soil particle which passes upon this bottom at the point x=().(), y = o.2, 2=9.5 is described by the following equations: z = 0.0000 1 622 (5^^ — 45.5^ + 342)^ + s (70) — 0.1 192-— 1.126x2+ 20.78%+ 10.032— 201.63 = (71) x^+ i.8>'^ — 42.41X— 1. 5_y+ 245.25 = (72) s = vt. (47) Jan. sS, 1918 Study of Plow Bottoms 171 From equations (70), (71), (72), and (47) the following are obtained: Table VIII. — Values (in inches) for — s z X y 18 27 36 iS. 4 27.4 36.0 7-55 19-5 3-6 8.25 II. 2=0.00001622(1;-/- — 45. 5'^/ +342 )^+7;f ^=o.oooo3244[{V-Y--45.5T;/+342)(2i'-/-45.5'z;)j+c; ^=o.oooo3244[(T;Y--45-S'^^+342)(2i;-)4-(2^'-f-45-5'"^Fl dx dt'' (. 2382+1. 126X-10.03) dH df^' 2X — 1. 1262 + 20. 78 2a;— 1. 1262+20. 78 dt 3. 63'-!. 5 dt- 3. 6}' -1. 5 The plow moved forward with a velocity of 36 inches per second, , .^=36/ From equations (74), (75), (76), {77), (78), (79), and (80) the listed in Table IX are computed. Table IX. — Values for — (73) (74) (75) (76) (77) (78) (79) giving (80) values dz dH dv d^-'v dz dh s / di dfi dt dt" dt df- .Sec. iS H 7.09 53-6 16. g 28. 4 37-7 - 9-07 27 H 25- 15 47-75 17-32 -50.0 34-44 — 10. 21 36 I 38-4 41. 6 3-44 -74-8 36 29.52 By making the proper substitutions from (80), (74), (76), and (78) in equations (54), (57), (58), and (59) the values of the direction cosines for the normals N the tangents to the path T, and the perpendiculars to the planes formed by the normals and tangents R for three points are computed and listed in Table X. 172 Journal of Agricultural Research Vol. XII, No. 4 Table X. — Values of the direction cosines for normals, tangents to the path, and per- pendiculars to the planes X=7-55 y=3(> 2=18.4 COS 7Vx= cos I^j= COS /v j= - 0-549 .716 - -429 cos T35= 0. 169 cosTy= .4025 cos T,= . 9 cos 2?x= cos Ry= cos R^= 0.817 - • 564 .0977 X=ll.5 J^8.2S 2=27-4 COS N^= cos A^y = COS Nj= - 0. 728 . 229 - .646 cos T'x=o. 546 cos Ty= . 376 cos T^= . 749 cos R^= cos Ry= cos /?j= 0. 4145 - -897 • 149 1=19.5 >■=" 2=36 COS iVx= COS A^y= - COS N^=- 0.698 - -215 - -683 cos T^=o. 728 cos Ty= . 065 cos Te= . 683 cos R^= cos Ry = cos /?j= 0. 102 - -975 . 2 For the purpose of computing the forces a block of soil 2 inches wide, 1 inch long, and }4 inch thick is taken. The mass of this soil is 1728.32.2. 12 32.2.12 p= density. (81) By the proper substitutions from Tables IX and X into equations (60) , (61), and (62) the forces necessary to produce the accelerations are com- puted and listed in Table XI. Table XI. — Forces necessary to produce acceleration in soil particles x= 7-55 y= 3-6 2=18.4 Fn= .00503P F'r= .001 1 6p Fr= .00252P x=ii.s >'= 8-25 2=27.9 Fn= .002S1P Ft= .000248P Fa= .00592P *=i9-5 y=ii 2=36 Fn= .00234P Ft= .oo4c8p Fe= .00778P A soil particle in passing over the surface of the plow bottom will be acted upon by the following : (a) A force from the surface of the bottom acting in the direction of the normal. (b) Gravity. (c) Pressure from the weight of the soil above the particle. (d) Friction between the particle and the surface. Jan. 28. 1918 Study of Plow Bottoms 173 (e) Shearing, stretching, or compression on each of the remaining five sides of the particle, due to its contact with other soil particles. The force which produces the movement of a soil particle in any direc- tion will be the resultant of the components of the above-listed forces which act in the direction of the movement. The preceding analysis of the motion which certain soil particles have in the operation of plowing has not been developed from as refined methods nor as uniform data in all cases as could be desired, but the re- sults obtained furnish abundant evidence that the problem here at- tempted is by no means hopeless. The study should be continued upon a tough sod, which would stretch more uniformly, and some apparatus which would remove the necessity of certain soil particles remaining in line with each other should be substituted for the pins. HISTORY OF THE DEVELOPMENT OF PLOW BOTTOMS The Annual Report of the New York State Agricultural Society for 1867 contains an ex- cellent treatise giving the geometrical con- struction of the sur- faces of many histori- cal plow bottoms, but no attempt has been made in that report to classify these SUr- ^^^ ,r.^^epcrrUrSro,e.,r,c3oc,.ey faces upon the basis ^^^ ^^ of their mathematical forms. Using the above-mentioned work as a basis, the author has attempted to work out the mathematical forms of the most important of these historical surfaces with a view to making fundamental compari- sons with present-day plow bottoms. JEFFERSON'S PLOW BOTTOM In 1788 Thomas Jefferson, while making a tour in Germany, devel- oped what appears to be one of the first methods recorded for making the surface of the moldboard geometrically exact in form.^ He argued that the offices of the moldboard were to receive the soil from the share and invert it with the least possible resistance. In order to do this, Jeffer- son developed a surface which he considered best adapted for the work of plowing, but attention should be called to the fact that no evidence is offered to prove the assertion. Figure 15 shows the framework for generating the Jefferson moldboard, in which lines em and oh are the directrices. To generate the surface a straightedge is laid upon eo and > GouiD, J. S., et al. Op. cit.. p. 403. 174 Journal of Agricultural Research voi. xii. No. 4 moved backward, the straightedge remaining parallel to the plane z^O. By taking the point o as the origin, the equation of the surface is ^by2—2dx2—2bly + 2bdz = 0^ (82) b = breadth of furrow (i= depth of furrow / = length of moldboard. On rotating the XV-axes through tan~^= 2C//36, the equation is {gF- + 4(P)y'z- ^bdlx' - 6bHy' + 2bd^gl^+4dFz= 0. (83) On rotating the "K'Z-axes through tan"^^^^^, the equation is (96^+ 4(i^) [(y"y-{z'y]-Sbdlx' + 2{bd-y/iSb^ + 8d'-3bH-yl2){y" + z'] = 0. (84) Translating the axes to the points y" = y"+yo Z' = Z" +Zo where yo has such a value that 2(9b' + 4d:')yo+2[bd-y/iSb'+8d?-3bH^/^] = 0, (85) and Zq has such a value that - 2i9b^-\- 4dr-)Zo+ 2[bd^TW+8d^- 3bH^] = 0, (86) gives {gF^-^4d')[(y''y-(z'y]-8bdlx^+i^-Zo')(9b' + 4d') + (yo+z,)(2bd^i8b' + Sd'-3bH^) = 0. (87) Letting the constant terms in (87) equal C gives (9b^ + 4d')[(y"y- (2'y-]-8bdlx'+C=0. (88) Translating the axes to the point x' = x"-}-Xo where Xq has such a value that -8bdlxo +C = gives {gb^-\-4d')[{y"y- {z'y] = 8bdlx". (89) This is the equation of a hyperbolic paraboloid.' LAMBRUSCHINl'S PLOW BOTTOM Lambruschini," an Italian, describes a method for generating the sur- face of a plow bottom which he considered to be more efficient than the surface developed by the Jefferson method. Lambruschini proposed ' The method of developing the equation for this surface is given upwn pages 150 to isd- * Snyder, Virgil, and Sisam, C. H. Op. cit., p. 73. 5 IvAMBRUScmNi, R. Op. cit., p. 37-80. 1832. Jan. 2S, 1918 Study of Plow Bottoms 175 a helacoid generated as follows: Lay out a rectangle opan (fig. 16) twice the desired width of the furrow and of an empirically determined length. Take the point m midway between points o and p and draw the line mm parallel to pq. A straightedge laid upon mo and moved backward along the line mrrii being kept parallel to the plane z = 0, and with an angular rotation pro- portional to the move- ment toward Wj, gen- erates the surface of the Lambruschini bot- tom. The point of the straightedge which was at a will describe the helix oo^q (fig. 16). The equation of this surface is y — = tan 6, X where ^ has uniformly increasing values as z increases. n Fig. 16. Then d = f (z), when 0=90 I = length of line mmi radians, I n_ I 2 2 n Hence, ='Kr)- (90) small's plow bottom ^ About 1760, a Scotchman, James Small, established a factory in Scotland for the manufacture of plows. The surface of Small's mold- ^ board is obtained by laying a straightedge upon op (fig. 17) and moving it backward parallel to the plane 2 = 0, with the line pm and the curve oh as directrices. The equa- tion of the curve, a half catenary, is obtained by drawing a line og (fig. 18) the length of line og (fig. 17). At o erect a line 00^ perpendicular to Hne og and equal in length to Hne gh (fig. 17). ' Gould, J. S., et al. Op. cit., p. 415. /"rom Report of NXStole Agnc.5oc l$S7 Fig. 17. 176 Journal of Agricultural Research Vol. XII. No. 4 Through point o^ (fig. 18) draw a hne oji parallel and equal to line og. With h and as points of suspension describe a catenary with its lowest point at O. Taking the point O (fig. 18) as origin, the equation of the catenary is ^ 1/= — (g21z/3ba_j_^— 21z/3ba\ foi) a = Og. Transferring the origin to the point a gives y= — (e-'^ ''^^^ -\- e~2'^ 1"^^^) — a (92) as the equation of the catenary oh (fig. 17). The equations of line pm (fig. 17) are y=0 x=h. h Fig. j8. Any plane parallel to the plane 2 = is given by z^c, and this plane cuts the line pm at the point y, = z^ = c. It also cuts the catenary oh at the point _3b Z2 = C. The equation of the line in the plane z = c which cuts the line pm and the catenary oh (fig. 17) is x—h _ y—0 (93) or 3* _j l(c)-0 2/ (%-6)/(c)-:^^c-6)=0 (94) Jan. 28, 1918 Stvdy of Plow Bottoms 177 As this line is always parallel to the plane z = 0, it follows that c=^z and m=fiz). From equations (92) and (94) then, («- &)rf- (e^'^/^*'* + e--'^'^ba) _ ^1 y?^c - 6^ = O, (95) which is the equation of Small's moldboard. STEPHEN'S PLOW BOTTOM * About the same time that Small brought out his moldboard another Scotchman named Stephens developed a method for forming the surface ^p^^ /nj/r? Reporr ofN YStoti Agric 3oc 1867 Fig. iq. From ffeport of NY Slate A! falls upon p, my upon m, and h^ upon h. This will locate the curve ph (fig. 19), leaving a figure as shown in figure 21. It will be »Gotn,D, J. S., et al. Op. cit., p. 431. 178 Journal of Agricultural Research Vol. XII. No. 4 observed in figure 2 1 that - = tan 6 where d has gradually increased values from O at 2 = to 90° at 2 = /. Further, 6=-r, radians where 7 represents the lengths of arcs 11', 22', etc.; then — = tan ( v )• From figure 20 the equation of the circle with its center at O, taking p^ as the origin is In figure 20 F=2nb cos «^+ 7= 90°; B+B'== 90°; a + B'+ 7=180°; '.^P + n'b^ ..b (98) nb^4n^b^-(^-^+'^) J_ 2nb' (99) Substituting the values for F from equation (98) and for G from equation (99) gives (100) ^ = tan[/(2)]. which is the equation of the surface. rahm's pi,ow bottom * In 1846 Rev. W. L. Rham, an Englishman, brought fonvard the theory that the lines of the moldboard running in the longitudinal direction should be straight, but that the section of the mold- board formed by any plane z = c (fig. 22) should be a straight line or a cur\^e, ac- cording to the phys- ical characteristics of the soil to be worked. Mr. Rham agreed that for medium, mellow soils the surface of the moldboard should be From Repwrt of N. Y. State Agric. Soc. 1867. Fig. 22. ' Gould, J. S., et al. Op. cit., p. 442. Jan. 28, 1918 Study of Plow Bottoms 179 generated by laying a straightedge upon oe and moving it backward parallel to the plane z = with the lines eji and em as directrices. This surface will be a portion of a hyperbolic paraboloid, the same general type as the surface which Jefferson proposed. The orthogonal projection of the generator in various positions upon the plane ^=0 will look as shown in figure 23. For stiff, clay soils the lines (fig. 24) e o From Report of N. Y. State Agric. Soc. 1867 Fig. 23. e o From Report of N. Y. State Agric. Soc. 1867 Fig. 24 e o From Report of N. Y. State Agric. Soc. 1S67 Fig. 2-. are made concave and for loose, sandy soils (fig. 25) they are made convex. As no exact description was given regarding the shape of the curves (fig. 24, 25), it has not been possible to develop equations for the surfaces. However, as it is known that these surfaces have straight lines in one direction and can not be described by an equation of the second order, they are of the fourth order or higher. KNOX's PLOW BOTTOM* In 1852 Samuel A. Knox, of Worcester, Mass., applied for a patent upon the surface of a plow bottom which was certainly unique. The skeleton of this surface is shown in figure 26. The seg- ments of circles I, II, and III are placed in parallel planes 12 inches apart, so that a series of straight lines will cut the three cir- cles. Circles I and III have equal diameters and the diameter of circle II is one-half that of circles I and III. As the equation of this surface is of the eighth order, it will not be worked out in detail, but a development will be given to show how the equation could be obtained. Let the equation of the three circles be ^ ^2 + ^2 = iv*2 z=0, z=k: from ReponoflitrSlaieAgricSoc /ii67 Fig. 1 Gould. J. S., et al. Op. cit., p. 49.';- • This development is the work of Virgil Snyder, Professor of Mathematics, Cornell University. i8o Journal of Agricultural Research voi. xii. No. 4 and {%-€)■+ {y-d)- = R- Z=2k. Draw the line from a point {x^, y\, O) on the first circle to a point {x^, y2, 2k) on the third. Its equations are x-x^_ y-y\ from which ^2~^i y2~>'i 2k 2fe(:x: — Xi)+2(Xj — c) ■ = Xo — c, 2k{y-y^^rz{y^-d) _ __ ^,_^ _ ^ _ Since {x,-cf^{y,-df = R}, we have, after simplifying, 4^'[(^-^i)'+(y->'i)']+4M(«-^i)(^i-c) ■^{y-yd{y-d)\^-z\{x,-cf+{y,-df-w\=o. (loi) This is the equation of a cone with vertex at {Xy, y^, O) and passing through the third circle. In the same way, find the equations of the line from (x^, y^ O,) to (3C3, ^3, k) on the middle circle k(x — x.) + z(x. — a) ' -x,-a, Hy-yi) + Hyi-b) _ _ ^ z " Since we have, after simplifying, k\{x-x,f+ {,y-y,Y\^ 2kz\_{x-x,){x,-a) ^{.y-h){y-y,)\^z^x-af^{y,~hf-~^ = 0. (102) When equations (loi) and (102) are multiplied out, it will be seen that x^i, y\ always enter in the form 7? ^-\- -f ^^^ R} . By substituting R? for *'i + Fi in each, the equations are of the form Axy + By^^C, A'x,+ B'y, = C'. Jan. j8, i9i8 Stiidy oj PI 07V Bottoms i8i Solve these equations for x^, y^ and put their values in x\ + f, = R\ A = [^kz{x+ c) — 2cz^ — 8a;P], B = \\hz\y -^d)- 2dz' - 8>//e2], C = [4i?'fe2 _ 4^2 (^2 + y2^ _ ^J^^^ _ ^^^y _ ^J^J^2^ ^ ^2 (^2 ^ ^2) J A ' = [2kz(x + a) — 2xk^— 2az'], B'=^[2kz{y+b) - 2y¥- 2hz% C = [R^k'' + k\x^ + y'')- ^kz{ax + hy- R^) + z\ct' + b' + ^R'-)l _B' C-BC ' ^'~AB'-A'B' _ C'A-CA' ^'' AB'-A'B' hence {B'C-BC'y+{C'A-CA'f = R\AB'-A'Bf. (103) CYLINDRICAL PLOW BOTTOMS In 1854 an American, Joshua Gibbs/ patented a plow bottom the surface of which is a portion of a circular cylinder. Taking a point upon the axis of the cylinder as the origin, the equation of this surface is -2 + -,- 1=0 (104) In some foreign countries, notably Germany, the hyperbolic cylinder has been suggested as suitable for forming the surface of the moldboard. In this connection it is interesting to note that any cylindrical surface can be described by an equation of the general form. mead's plow bottom ^ In 1863 a Mr. Mead, of New Haven, Conn., patented a plow bottom, the surface of which conformed exactly to a portion of a frustrinn of a cone. The general equation of this surface is J2 /1/2 .5,2 -2 + T2-~2-0 (106) a^ P c^ holbrook's plow bottom The Report of the New York State Agricultural vSociety for 1867 con- tains a very complete report of the plow trials held at Utica, N. Y., in 1867, at which trials a line of plows designed by F. F. Holbrook, of Boston, Mass., showed general superiority to all other makes. The 1 Gould, J. S., et al. Op. cit., p. 502. ^ gquld, J. S., et al. Op. cit., p. 505. • Snyder, Virgil, and Sisam, C. H. Op. cit., p. 83. 1 82 Journal of Agricultural Research voi. xii. no. 4 following quotation gives a very good description of the Holbrook sur- faces : We ' were interested in the most minute details of these plows by Gov. Holbrook and the trials at Utica and subsequently at Brattleboro, Vt., showed very clearly the influence of the warped surface which is generated by his method upon the texture of the soil. Gov. Holbrook is as yet unprotected by a patent on his method, and we are therefore most reluctantly compelled to withhold a description of it but we have no hesitation in saying that it is the best system for generating the true curve of the moldboard which has been brought to our knowledge. This method is applicable to the most diversified forms of plows, to long or short, to broad or narrow, to high or low, no matter what the fonn may be, this method will impress a family likeness upon them all. There will be straight lines in each running from the front to the rear and from the sole to the upper parts of the share and moldboard. None of these lines will be parallel to each other, nor will any of them be radii from a common cen- ter. The angle formed by any two of them will be unlike the angle formed by any other two; a change in the angle formed by any transverse lines will produce a corresponding change in the vertical lines, and there will always, in every form of this plow, be a reciprocal relation between the transverse and vertical ^ lines. Plows made upon this plan may appear to the eye to be as widely different as it is possible to make them, and yet, on the application of the straightedge and protractor, it will be found that they agree precisely in their fundamental character. The stirface of the moldboard is always such that the different parts of the fiu-row slice will move over it with unequal velocities. From the above description it is evident that the surfaces of the Hol- brook plows are portions of a hyperboloid of one sheet whose general equation is — jL.y' ——— MISCELLANEOUS PLOW BOTTOMS In addition to the surfaces already described there remain at least three which show unique characteristics, but data were not available for developing the equations. In I Si 8 Gideon Davis,^ of Maryland, patented the surface of a plow bottom which was obtained by using the segment of a circle as a gen- erator and two segmerits of another circle as directrices. Somewhat later, 1834, James Jacobs,^ another American, brought out a plow bottom the surface of which was a combination of two mathematical surfaces, each of which had sets of straight lines in two directions. In 1839 Samuel Witherow, of Gettysburg, Pa., and David Pierce, of Philadelphia, Pa., brought out a plow bottom whose surface was gen- erated by the most ingenious use of the arc of a cycloid. A more detailed description of this plow can be found in the Report of the New York State Agricultural Society for 1867.^ • Gould, J. S., et al. Op. dt., p. 586. ' It should be noted that the lines here called transverse are designated as longitudinal (PI. 2, A), and the lines called vertical are designated as transverse. ' Gould, J. S., et al. Op. cit., p. 432. * Idem, p. 486. *Idem, p. 491. PLATE 6 A. — A plow bottom with two sets of straight lines. B. — A plow bottom, the surface of which is composed of each of two surfaces. C. — A plow bottom similar to B, but with the surfaces merging into each other farther back on the moldboard. D. — A plow bottom, the surface of which does not contain an infinite set of straight lines. Study of Plow Bottoms Plate 6 Journal of Agricultural Research Vci.XII, Nu. 4 study of Plow Bottoms Plate 7 '"Lj£jii" Journal of Agricultural Research Vol. XII, No. 4 PLATE 7 A —A plow bottom with a convex surface which has two sets of straight lines. B.-Instrument for measuring the space coordinates of any point of the plow bottom. C._A sod plow showing the furrow slice turned by it. PLATE 8 A. — Rows of wooden pins driven into the sod for estimating the stretch of the fur- row slice. B. — Furrow slice showing the position of the pins when on the moldboard. study of Plow Bottoms Plate 8 rJ i *