Eng. G. D. Terskov.

On the Construction of Moldboards for American Plows

1. Introduction.

The investigation of American moldboards carried out in Wiskhom by Prof. N. D. Luchinskii showed that their principal difference from moldboards of the European type consists in the arrangement of the rectilinear generators.

Whereas in European moldboards the rectilinear generators are arranged parallel to the bottom of the furrow and are inclined only toward the wall of the furrow, in American moldboards these generators are inclined both toward the bottom of the furrow and toward its wall (see Fig. No. 1).

As one moves upward, in European moldboards the inclination of the generators changes only with respect to the wall of the furrow (with respect to the bottom they remain parallel over the whole moldboard); in American moldboards, however, as one moves upward, the inclination of the generators changes both with respect to the bottom and to the wall of the furrow.

These investigations also showed that the position of the generators on the moldboard may be characterized by means of the so-called orthogonal curve, i.e., a curved line drawn through the right-hand end of the plowshare in such a way that this curve makes an angle equal to 90° with all the generators. In the majority of the moldboards studied, this orthogonal curve proved to be plane, and in shape approaching an ellipse.

European moldboard

American moldboard

Fig. 1.

In order to construct a moldboard of the American type, it proves necessary to have data:

a) on the position of the share relative to the bottom and wall of the furrow,
b) on the position of the plane of the orthogonal curve relative to the edge of the share and the bottom of the furrow,
c) on the position of the orthogonal curve,
d) on the form of the orthogonal curve itself, and
e) on the position of the generators relative to the plane of the orthogonal curve.

Judging from the published material (see N. D. Luchinskii, Investigation of American Tractor Bodies), the magnitude of these unknowns and their connection with one another and with the character of the soil being worked have not yet been clarified to such an extent that an American moldboard could be designed without turning the matter into a blind copying of existing models.

The inevitable inaccuracies in measurements of existing moldboards lead to the result that, even after taking all the necessary dimensions from nature, it is almost impossible to obtain a moldboard similar to the existing one, owing to various inconsistencies.

Below are given some considerations on the mutual relation between the quantities characterizing the form of American moldboards.

Fig. 2.

2. Relation between the position of the share and the orthogonal curve.

Let in Fig. No. 2 the plane (I) represent a vertical plane drawn perpendicular to the edge of the share; with the bottom of the furrow this plane intersects along the horizontal line “\(BP\),” perpendicular to the edge of the share “\(BA\),” and with the moldboard along some curve “\(B1\).”

If the point “\(B\)” is located at the right end of the edge of the share, then through it there will also pass the plane of the orthogonal curve (II), intersecting the bottom of the furrow along the line “\(BO\),” and the moldboard along the orthogonal curve “\(B2\).”

Since the curves “\(B1\)” and “\(B2\)” at the point “\(B\)” are perpendicular to the blade of the share and merge with one another, the intersection of planes I and II will be along a certain line “\(BR\),” tangent at the point “\(B\)” to both curves and lying above the line “\(BP\).” Therefore the angle “\(RBP\)” will obviously characterize the magnitude of the inclination of the share to the bottom of the furrow at the place of their contact, i.e., it will be the so-called cutting angle. We shall denote it by “\(\gamma\).” In turn, the angle “\(RBO\)” will characterize the position of the orthogonal curve with respect to the horizontal trace of the plane of the orthogonal curve (II). In other words, it will be the angle of inclination of the tangent to the orthogonal curve, at the point “\(B\),” with respect to the horizontal trace of its plane. We shall denote this angle by “\(\lambda\).”

The angle between the normal to the share “\(BP\)” and the horizontal trace of the plane of the orthogonal curve “\(BO\),” measured in the plane of the bottom of the furrow, we shall denote by “\(\sigma\).”

Finally, let us draw one more plane (III), perpendicular to the horizontal trace of the plane of the orthogonal curve. This plane, being vertical, will intersect plane (I) along the vertical line “\(PR\),” plane (II) along the inclined line “\(OR\),” and the bottom of the furrow along the horizontal line “\(OP\).” Obviously the angle “\(POR\)” will characterize the inclination of the plane of the orthogonal curve to the bottom of the furrow; we shall denote it by “\(\psi\).” The angles “\(BOR\),” “\(BOP\),” “\(OPR\),” and “\(BPR\)” will obviously be right angles.

The resulting spatial quadrilateral “\(BOPR\)” makes it possible to write the following relations:

\[ PR = PO\,\operatorname{Tg}\psi = PB\,\operatorname{Tg}\gamma \tag{1} \]

\[ PO = PB\,\operatorname{Sn}\sigma \tag{2} \]

\[ PO = OR\,\operatorname{Cos}\psi = OB\,\operatorname{Tg}\sigma \tag{3} \]

\[ OR = OB\,\operatorname{Tg}\lambda \tag{4} \]

From (1) we find

\[ \operatorname{Tg}\psi = \frac{PB}{PO}\,\operatorname{Tg}\gamma \tag{1-a} \]

From (2) we find

\[ \frac{PB}{PO} = \frac{1}{\operatorname{Sn}\sigma} \tag{2-a} \]

Substituting (2-a) into (1-a), we obtain

\[ \boxed{\operatorname{Tg}\psi = \frac{\operatorname{Tg}\gamma}{\operatorname{Sn}\sigma}} \tag{5} \]

From (3) we find

\[ \operatorname{Cos}\psi=\frac{OB}{OR}\operatorname{Tg}\sigma \tag{3-a} \]

From (4) we find

\[ \frac{OB}{OR}=\frac{1}{\operatorname{Tg}\lambda} \tag{4-a} \]

Substituting (4-a) into (3-a), we obtain

\[ \boxed{\operatorname{Cos}\psi=\frac{\operatorname{Tg}\sigma}{\operatorname{Tg}\lambda}} \tag{6} \]

Moreover, eliminating the angle “\(\psi\)” from (5) and (6), we find

\[ \boxed{\operatorname{Cos}\sigma=\frac{\operatorname{Cos}\lambda}{\operatorname{Cos}\gamma}} \tag{7} \]

Since the angles “\(\lambda\)” and “\(\gamma\)” and their cosines are essentially positive, and \(\operatorname{Cos}\sigma\) cannot be greater than unity, it is evidently possible to write the inequality

\[ \frac{\operatorname{Cos}\lambda}{\operatorname{Cos}\gamma}\leq 1 \]

whence

\[ \operatorname{Cos}\lambda \leq \operatorname{Cos}\gamma \]

or

\[ \lambda \geq \gamma \tag{8} \]

i.e. the angle of inclination of the orthogonal curve at the point “\(B\)” to the horizontal trace of its plane cannot be less than the cutting angle.

In the limiting case, if

\[ \lambda=\gamma \]

then from formula (7) we have

\[ \operatorname{Cos}\sigma=1;\qquad \sigma=0^\circ \]

from formula (6) we have

\[ \operatorname{Cos}\psi=0 \qquad \psi=\frac{\pi}{2} \]

i.e.—“If the angle of inclination of the orthogonal curve, at the point ‘\(B\),’ to the horizontal trace of its plane is equal to the cutting angle, then the plane of the orthogonal curve becomes perpendicular to the bottom of the furrow and to the blade of the plowshare”—in other words, in Fig. No. 2, plane (II) will merge with plane (I), and the line “\(OB\)” with the line “\(PB\),” and in

as a result, the guiding curve will be obtained in the form in which it is specified when designing European moldboards by the method of Prof. V. P. Goryachkin.

3. Relation between the position of the lower generatrix and the orthogonal curve.

Let in Fig. No. 3, “\(OA\)” represent the wall of the furrow, \(AB\)—the blade of the share, and “\(OB\)”—the horizontal trace of the plane of the orthogonal curve.

Fig. 3.

curve. If we take as the vertical plane of projection the vertical plane perpendicular to the horizontal trace of the plane of the orthogonal curve, and accordingly as the axis of projection the line “\(OM_0\)”—perpendicular to “\(OB\),” then the vertical trace of the plane of the orthogonal curve, “\(OO'\),” will be inclined to the axis of projection “\(OM_0\)” at an angle “\(\psi\),” characterizing the inclination of the plane of the orthogonal curve to the bottom of the furrow. The horizontal projection of the blade of the share will be “\(AB\),” and its vertical projection—the line “\(OE\).”

Let us rotate the moldboard about the horizontal trace “\(OB\)” counterclockwise through the angle “\(\psi\),” i.e., until the plane of the orthogonal-

… curve with the bottom of the furrow; in this case the vertical projection of the share blade will occupy the position “\(OE'\),” and the horizontal one the position “\(Bg\),” the orthogonal curve “\(B2\)” is projected onto the bottom of the furrow without distortion of its form, just as is the angle “\(\lambda\)” between the horizontal trace “\(OB\)” of the plane of the orthogonal curve and the tangent “\(BR\)” to the orthogonal curve at the point “\(B\).”

Since we now look at the plane of the orthogonal curve (in horizontal projection) from above downward, perpendicular to it, all generators in the horizontal projection must pass perpendicular to the orthogonal curve. And since, in a correctly projected moldboard, the lower generator must coincide with the share blade, it is obvious that the horizontal projection “\(Bm'_0\)” of this generator (perpendicular “\(BR\)”) must coincide with the horizontal projection of the share blade “\(Bg\),” while the vertical projection of this generator “\(OM'_0\)” will coincide with the vertical projection of the blade “\(OE'\).”

In connection with this we may write that

\[ Om'_0 = OM'_0 \operatorname{Cos}\psi = OM_0 \operatorname{Cos}\psi = OB \frac{\operatorname{Cos}\psi}{\operatorname{Tg}\sigma} \tag{9} \]

but

\[ Om'_0 = OB \frac{1}{\operatorname{Tg}\lambda} \tag{10} \]

therefore

\[ OB \frac{1}{\operatorname{Tg}\lambda} = OB \frac{\operatorname{Cos}\psi}{\operatorname{Tg}\sigma} \tag{11} \]

whence

\[ \boxed{\operatorname{Cos}\psi = \frac{\operatorname{Tg}\sigma}{\operatorname{Tg}\lambda}} \tag{6} \]

i.e. we obtain the relation already known to us.

In addition, the point \(M'_0\) is the vertical trace of the generator of the moldboard surface passing through the point “\(B\),” and therefore, if the angle of inclination of this generator to the plane of the orthogonal curve is denoted by “\(\alpha_0\)” (in our case, when the orthogonal curve is brought into coincidence with the bottom of the furrow, this angle may be regarded as the angle of inclination of the lower generator to the bottom of the furrow), then one may write the relation (see Fig. No. 4)

Fig. 4.

\[ Bm'_0 \operatorname{Tg}\alpha_0 = m'_0 M'_0 \tag{12} \]

but

\[ m'_0M'_0 = Om'_0 \operatorname{Tg}\psi = OB \frac{\operatorname{Tg}\psi}{\operatorname{Tg}\lambda} \tag{13} \]

and

\[ Bm'_0 = \frac{OB}{\operatorname{Sn}\lambda} \tag{14} \]

therefore

\[ OB \frac{\operatorname{Tg}\alpha_0}{\operatorname{Sn}\lambda} = OB \frac{\operatorname{Tg}\psi}{\operatorname{Tg}\lambda} \tag{15} \]

whence

\[ \boxed{\operatorname{Tg}\alpha_0 = \operatorname{Tg}\psi \operatorname{Cos}\lambda} \tag{16} \]

In the extreme case considered above, when \(\lambda=\gamma\) and \(\psi=\dfrac{\pi}{2}\),

\[ \operatorname{Tg}\alpha_0 = \infty \quad \alpha_0=\frac{\pi}{2} \]

i.e. the lower generatrix (coinciding with the blade of the share) is situated with respect to the plane of the orthogonal curve at an angle of \(90^\circ\).

4. Shape of the Orthogonal Curve.

According to the investigations of N. D. Luchinskii, the shape of the orthogonal curve is closest to an ellipse. However, for example, for the plow of “Rostselmash” (John-Deere No. 6c), the parabola \(y^2=506x\) is the closest fit. Likewise, for example, in plows \(SP\ 212\) and M. Harris investigated by Prof. N. D. Luchinskii, a parabola will suit the orthogonal curve with no less accuracy than an ellipse.

In any case, one must admit that both the parabola and the ellipse can be the orthogonal curve. The parabola has, as compared with the ellipse, the advantage that its equation is much simpler.

Thus, for example, if in Fig. No. 5 the point “\(B\)” is the right end of the share, and the line “\(BO\)” is the horizontal trace of the plane of the orthogonal curve, then, denoting the coordinates of the point “\(B\)” by

Fig. 5.

with respect to the coordinate axes associated with the parabola, \(x_b\) and \(y_b\), we obtain

\[ x_c = y_b \operatorname{Cos} E + x_b \operatorname{Sn} E \tag{17} \]

\[ y_c = y_b \operatorname{Sn} E - x_b \operatorname{Cos} E \tag{18} \]

whence, after the corresponding transformations, we find

\[ x_b = x_c \operatorname{Sn} E - y_c \operatorname{Cos} E \tag{19} \]

\[ y_b = y_c \operatorname{Sn} E + x_c \operatorname{Cos} E \tag{20} \]

The angle of inclination of the tangent to the parabola at the point “\(B\),” with respect to the axis of symmetry of the parabola, is determined from the expression

\[ \operatorname{Tg}\Delta = y^{1} = f^{1}(x_b) = \left|2px\right|_{x=x_b}^{1} = \frac{P}{y_b} \]

but

\[ \boxed{ P = \frac{y^{2}}{2x} = \frac{y_b^{2}}{2x_b} = \frac{(y_c \operatorname{Sn} E + x_c \operatorname{Cos} E)^{2}} {2(x_c \operatorname{Sn} E - y_c \operatorname{Cos} E)} } \tag{21} \]

Therefore

\[ \boxed{ \operatorname{Tg}\Delta = \frac{y_c \operatorname{Sn} E + x_c \operatorname{Cos} E} {2(x_c \operatorname{Sn} E - y_c \operatorname{Cos} E)} = \frac{y_c \operatorname{Tg} E + x_c} {2(x_c \operatorname{Tg} E - y_c)} } \tag{22} \]

Thus, if “\(x_c\),” “\(y_c\),” and “\(E\)” are given, i.e. the position of the vertex and the axis of symmetry of the parabola, then it is possible to find the magnitude of the angle “\(\Delta\)” and the parameter of the parabola “\(P\).”

In addition, directly from Fig. No. 5 it is evident that the angle of inclination of the tangent to the parabola at the point “\(B\)” with respect to the horizontal trace of the plane of the orthogonal curve is

\[ \boxed{ \lambda = E + \Delta - \frac{\pi}{2} } \tag{23} \]

The formulas derived here, (21), (22), (23), (7), (5), and (16), include 10 quantities:

\[ P,\ x_c,\ y_c,\ E,\ \Delta,\ \lambda,\ \gamma,\ \sigma,\ \psi,\ a_0 \]

In addition, in order to construct the moldboard it is necessary to know the angle of the share blade with the wall of the furrow, “\(\Sigma\),” and the law of variation of the angle of inclination of the generators to the plane of the orthogonal curve. Thus, in all, there are 12 quantities. Six of them can be determined from the equations given above; two, “\(\Sigma\)” and “\(\gamma\),” have already been established by practice. Four quantities still remain unknown—

quantities: \(E''\), \(y_c''\), \(x_c''\)—characterizing the position of the orthogonal curve, and the law of variation of the angle of inclination of the generators to the plane of the orthogonal curve. Therefore it is obvious that the design of the American moldboard can be carried out completely only when, as a result of the appropriate investigations, the values of the last four quantities and their relation to the nature of the soil being worked have been ascertained.

5. Construction of the moldboard.

From the material presented above it follows that, for the conscious construction of the surface of an American moldboard, it is necessary to know the form and position of the orthogonal curve, as well as the law of variation of the angle of inclination of the generators to the plane of the orthogonal curve. Let us take these from existing moldboards.

Suppose, for example, that the “Rostselmash” plow is taken as the original. As a result of measuring it we have the following data:

1) Cutting angle \(\gamma = 22^\circ\)

2) Angle of the share with the wall of the furrow \(\Sigma = 41^\circ\)

3) The orthogonal curve is a parabola with vertex determined by the coordinates \(x_c = 176\ \text{mm}\), \(y_c = 195\ \text{mm}\), and with an axis of symmetry located at an angle \(E = 73^\circ\) to the perpendicular dropped to the horizontal trace of the plane of the orthogonal curve.

4) The difference between the angles of inclination of the upper and lower generators to the plane of the orthogonal curve is \(\Delta\alpha = 25^\circ\), with the variation of the angles of inclination proceeding in such a way that the lower half of the moldboard accounts for

\[ \frac{1}{3}\Delta\alpha, \]

and the upper half for

\[ \frac{2}{3}\Delta\alpha. \]

a) Determination of the angles characterizing the moldboard.

By formula (21) we find the parameter of the parabola

\[ P=\frac{(195\,Sn\,73^\circ+176\,Cos\,73^\circ)^2}{2(176\,Sn\,73^\circ-195\,Cos\,73^\circ)}=253\ \text{mm} \]

Thus the equation of the orthogonal curve became

\[ y^2=506x \]

By formula (22) we find the angle of inclination of the tangent to the parabola at the point \(B''\), with respect to the axis of symmetry of the parabola,

\[ Tg.\ \Delta=\frac{195\,Tg\,73^\circ+176}{2(176\,Tg\,73^\circ-195)}=1{,}07;\qquad \Delta=47^\circ \]

By formula (23) we find the angle of inclination of the tangent to the parabola at the point \(B''\), with respect to the horizontal trace of the plane of the orthogonal curve.

\[ \lambda = 73^\circ + 47^\circ - 90^\circ = 30^\circ \]

By formula (7) we find the angle of inclination of the horizontal trace of the plane of the orthogonal curve with respect to the perpendicular erected to the share blade from its right end:

\[ \operatorname{Cos}\sigma = \frac{\operatorname{Cos}30^\circ}{\operatorname{Cos}22^\circ} = 0.933;\quad \sigma = 21^\circ \]

By formula (5) we find the angle of inclination of the plane of the orthogonal curve to the bottom of the furrow:

\[ \operatorname{Tg}\psi = \frac{\operatorname{Tg}22^\circ}{\operatorname{Sn}21^\circ} = 1.13;\quad \psi = 48^\circ 30' \]

By formula (16) we find the angle of inclination of the lower generator to the plane of the orthogonal curve.

\[ \operatorname{Tg}\alpha_0 = \operatorname{Tg}48^\circ 30' \operatorname{Cos}30^\circ = 0.98.\ \alpha_0 = 44^\circ 30' \]

b) Construction of the orthogonal curve (Fig. No. 6).

Taking the line “\(NN\)” as the furrow wall, at a certain point “\(A\)” on it we lay off the angle of the share with the furrow wall \(\Sigma = 41^\circ\) and construct the share blade

\[ AB' = \frac{b}{\operatorname{Sn}\Sigma} = \frac{300}{\operatorname{Sn}41^\circ} = 455\ \text{mm} \]

Through the right end of the share “\(B\)” we draw the horizontal trace of the plane of the orthogonal curve, “\(OB\),” at an angle \(\sigma = 21^\circ\) to the perpendicular erected from point “\(B\)” of the share blade. Through the point “\(O\)” of intersection of the horizontal trace of the plane of the orthogonal curve with the furrow wall, we draw the line “\(MM\),” perpendicular to the trace “\(OB\),” and the vertical plane passing through the line “\(MM\)” is taken as the vertical plane of projections. We find the projection of the share on the axis of projection “\(OE\).”

We rotate the moldboard surface about the horizontal trace “\(OB\)” counterclockwise until the plane of the orthogonal curve coincides with the bottom of the furrow; in doing so, the vertical projection of the share blade “\(OE\)” will turn through the angle \(\psi = 48^\circ 30'\) and take the position “\(OE'\),” while the horizontal projection will move into the position “\(Bg\)” (with the nose of the share moving into the position “\(g\),” moving parallel to the line “\(MM\)” until it lies on the same direction of projection as point “\(E'\)”).

Since the plane of the orthogonal curve has coincided with the bottom of the furrow, it is evident that the orthogonal curve will be visible on the bottom of the furrow in undistorted form and therefore can be drawn from the known data.

\[ x_c = 176\ \text{mm};\quad y_c = 195\ \text{mm};\quad y^2 = 506x;\quad E = 73^\circ \]

For the work of Eng. G. D. Terskov

Construction of an American Moldboard of the Universal Type

Scale 1:10

Fig. No. 6

Labels in the drawing:

• Horizontal trace of the control plane.
• Vertical trace of the field moldboard.
• Vertical trace of the field moldboard; surface set with the bottom of the furrow from the breast.
• Projection of the edge slope onto the furrow wall.
• Vertical trace of the field moldboard.
• Projection of the moldboard curve onto the furrow wall.
• Furrow surface.
• Moldboard curve on the bottom of the furrow.
• Projection of the upper trace onto the furrow wall.

by points or graphically. In constructing the orthogonal curve it must be borne in mind that, if the construction has been carried out correctly, the parabola will pass exactly through the point “\(B\).” On the constructed orthogonal curve, beginning from the point “\(B\),” we take, at equal distances from one another, a series of points I, II, III . . . . . IX, and find the vertical projections of these points—they will evidently lie on the line “\(MM\)” in the corresponding directions of projection. Let us denote these projections by

\[ \mathrm{I}'_{0},\ \mathrm{II}'_{0},\ \mathrm{III}'_{0}\ .\ .\ .\ .\ .\ \mathrm{IX}'_{0} \]

We rotate the plane of the orthogonal curve about the horizontal trace “\(OB\)” clockwise through the angle \(\psi = 48^\circ 30'\)—i.e., into its normal position. In doing so, the vertical projections of the points of the orthogonal curve will describe arcs equal to “\(\psi\)” and will be arranged along a straight line (at the angle “\(\psi\)” to the line “\(MM\)”), while the horizontal projections will move in a direction parallel to the line “\(MM\),” until they lie on the same directions of projection as the new positions of the vertical projections. Connecting the points thus obtained

\[ 0,\ 1,\ 2,\ 3,\ .\ .\ .\ .\ .\ 9 \]

by a smooth curve, we obtain the horizontal projection of the orthogonal curve in its normal position.

To find the projection of the points of the orthogonal curve on the furrow wall, from the points \(0, 1, 2, . . . . . 9\) of the horizontal projection we draw directions of projection perpendicular to the line “\(NN\)” and set off on them, above the line “\(NN\),” segments equal to the distances of the vertical projections from the line “\(MM\).” Connecting the points thus obtained \(0', 1', 2', 3', . . . . 9'\) by a smooth curve, we obtain the projection of the orthogonal curve on the furrow wall.

The last step in constructing the orthogonal curve consists in finding the profile projection of the orthogonal curve, and presents no difficulties—the determination is carried out according to the rules of descriptive geometry. As a result we find the points

\[ 0'',\ 1'',\ 2'',\ 3'',\ .\ .\ .\ .\ 9'' \]

and, connecting them by a smooth curve, we obtain the profile projection of the orthogonal curve.

c) Construction of the profile projection of the moldboard.

This construction is carried out in the usual way (see Prof. Sladkov, Graphical Method for Constructing Plow Bodies), taking into account the known data on American moldboards (see Agricultural Machine, 1932, No. 4, p. 21).

c) Determination of the angles of inclination of the generators to the plane of the orthogonal curve.

Laying off along the axis of abscissas, at equal distances from one another, the points 0, 1, 2, 3, . . . 9 (Fig. No. 7); at points 0, 3, and 7 we lay off ordinates representing, on a certain scale, the angles

Fig. 7.

\[ \alpha_0 = 44^\circ 30' \]

\[ \alpha_3 = \alpha_0 + \frac{1}{3}\Delta\alpha \simeq 52^\circ \]

\[ \alpha_7 = \alpha_0 + \Delta\alpha = 70^\circ \]

Through the ends of these ordinates we draw a smooth curve, from which we also find the magnitudes of the angles for the remaining points:

\[ \alpha_1 = 46^\circ 30';\quad \alpha_2 = 49^\circ;\quad \alpha_4 = 56^\circ;\quad \alpha_5 = 60^\circ;\quad \alpha_6 = 64^\circ 30';\quad \alpha_8 = 75^\circ;\quad \alpha_9 = 80^\circ \]

d) Finding the projections of the generators of the moldboard.

At the selected points 0, I, II, III, . . . IX of the orthogonal curve, by methods known from mathematics, we construct normals and extend them to their intersections, at the points \(m_0'\), \(m_1'\), \(m_2'\), . . . \(m_9'\), with the axis of projection “\(MM\).”

Obviously, the lines \(O—m_0'\), \(I—m_1'\), \(II—m_2'\), . . . \(IX—m_9'\) will be the horizontal projections of the generators of the moldboard when the plane of the orthogonal curve is made to coincide with the bottom of the furrow. If all the con—

If the constructions have been carried out correctly, then \(O—m_0'\) will coincide with the horizontal projection of the share blade \(Bg\).

The points of intersection of the generators with the vertical plane of projections

\[ M_0',\; M_1',\; M_2', \ldots M_9' \]

are found from the relation:

\[ M_0'm_0' = Om_0'\operatorname{Tg}\alpha_0.\quad M_1'm_1' = 1m_1'\operatorname{Tg}\alpha_1 \ldots\ldots M_9'm_9' = IX.\,m_9'\operatorname{Tg}\alpha_9 \]

or graphically, as is shown in Fig. No. 8.

Fig. 8.

By connecting these points with a smooth curve, we obtain the vertical trace of the moldboard surface when the plane of the orthogonal curve is brought into coincidence with the bottom of the furrow. In this connection it must be borne in mind that the correctness of the constructions is proved by the coincidence of the point \(M_0'\) with the line \(OE'\).

Next we find the position of the vertical trace of the moldboard surface for the normal position of the plane of the orthogonal curve.

For this purpose we rotate the curve \(M_0' M_1' M_2' \ldots M_9'\) about the point “\(O\)” clockwise through the angle “\(\psi\)” to the position \(M_0 M_1 M_2 \ldots M_9\); in this case the horizontal projections of the new positions of the vertical traces of the moldboard generators will be found on the line “\(MM\),” in the corresponding directions of projection; let us denote these projections by

\[ m_0,\ m_1,\ m_2,\ldots m_9 \]

Connecting “\(m_0\)” with “\(B\)”; “\(m_1\)” with “\(1\)”; “\(m_2\)” with “\(2\)” … “\(m_9\)” with “\(9\),” we obtain the horizontal projections of the moldboard generators.

To find the projection of the moldboard generators on the furrow wall, it is first necessary to find the projection of the vertical trace of the moldboard surface on the furrow wall, \(N_0\ N_1\ N_2 \ldots N_9\).

For this purpose, from the points \(m_0,\ m_1,\ m_2,\ldots m_9\) we drop perpendiculars to the line “\(NN\),” and from the points of their intersection with the line “\(NN\)” we set off upward along the perpendiculars segments equal to the heights of the points \(M_0,\ M_1,\ M_2,\ldots M_9\) above the line “\(MM\),” i.e.
\(n_0 N_0 = m_0 M_0;\ n_1 N_1 = m_1 M_1;\ldots n_9 N_9 = m_9 M_9.\)

Connecting “\(1'\)” with “\(N_1\),” “\(2'\)” with “\(N_2\),” …, “\(9'\)” with “\(N_9\),” we obtain the projections of the moldboard generators on the furrow wall.

To find the profile projections of the moldboard generators, it is necessary first to find the field edge of the moldboard, i.e. the curve of intersection of the moldboard surface with a vertical plane coinciding with the furrow wall. For this purpose we project the points of intersection of the horizontal projections of the moldboard generators with the line “\(NN\)” onto the corresponding projections of the generators on the furrow wall. The resulting points \(A_1',\ A_2',\ A_3',\ldots A_9'\) are connected by a smooth curve, which will be the field edge of the moldboard surface.

Transferring the points \(A_1',\ A_2',\ A_3',\ldots A_9'\) of the field edge to the profile projection of the moldboard, we obtain the points \(A_1'',\ A_2'',\ A_3'',\ldots A_9''\); connecting the latter with the corresponding points of the orthogonal curve, i.e. \(A_1''\) with \(1''\); \(A_2''\) with \(2''\); \(A_3''\) with \(3''\); … \(A_9''\) with \(9''\), we obtain the profile projections of the moldboard generators.

e) Construction of the vertical and horizontal projections of the moldboard.

This construction is carried out in the usual way: namely, the points of intersection of the generators with the profile projection of the moldboard are first transferred to the corresponding projections of the generators on the furrow wall, and from there, in an analogous manner, to the horizontal projections of the generators. Connecting the resulting points by smooth curves, we find the outline of the moldboard on the horizontal projection and on the furrow wall.