G. D. Terskov.

On the Design of the Cutting Apparatus of Harvesting Machines

1. Trajectory of Motion of an Arbitrary Point of the Knife.

Relative motion.

The knife of harvesting machines, with respect to the finger bar, performs a reciprocating motion. In order to characterize this motion completely, it is necessary to know the so-called stroke of the knife and, in addition, to have the equation of motion.

The stroke of the knife is the path traversed by any point of the knife from its extreme right position to its extreme left position; therefore its value (Fig. 1) will be equal to

\[ s = O_1B_2 - O_1B_1 = \sqrt{(e+r)^2-H^2} - \sqrt{(e-r)^2-H^2} = \]

\[ = (e+r)\sqrt{1+\left(\frac{H}{e+r}\right)^2} - (e-r)\sqrt{1-\left(\frac{H}{e-r}\right)^2}. \]

Fig. 1.

Since the second terms under the radicals, for harvesting machines, are in any case no greater than 0.15, then, applying the formula

\[ \sqrt{1\pm\delta}\simeq 1\pm\frac{\delta}{2}, \tag{1} \]

for the approximate extraction of the square root, after the corresponding transformations, with an error of not more than 1%, we obtain

\[ s = 2r\left[1+\frac{H^2}{2(e^2-r^2)}\right] \tag{2} \]

Here \(r^2\) is negligibly small in comparison with \(e^2\), and therefore it may be discarded without appreciable error; in that case

\[ \boxed{s=2r\left[1+\frac{H^2}{2e^2}\right]} \tag{2a} \]

To derive the equation of motion of any point of the knife, we shall take the extreme right-hand position as the initial one, and shall measure the angles of rotation of the crank from the right dead-center position; in that case

\[ x=A_0B-A_0O_1-O_1B_1=e\cos\psi-r\cos(\alpha_1+\alpha)-(e-r)\cos\alpha_1. \]

In this expression \(\psi\) is a variable quantity and depends on the position of the crank. From Figs. \(ABA_0\) and \(AOO_1A_0\) we have

\[ l\sin\psi=H+r\sin(\alpha_1+\alpha), \]

and therefore

\[ \sin\psi=\frac{H+r\sin(\alpha_1+\alpha)}{e}. \tag{3} \]

\[ \cos\psi=\sqrt{\,1-\left[\frac{H+r\sin(\alpha_1+\alpha)}{e}\right]^2\,} \tag{4} \]

Extracting the square root, from formula (1) we shall have

\[ \cos\psi \cong 1-\frac{[H+r\sin(\alpha_1+\alpha)]^2}{2e^2} =1-\frac{H^2}{2e^2}-\frac{Hr\sin(\alpha_1+\alpha)}{e^2} \tag{4a} \]

The last term is very small, and therefore it may quite well be neglected, with an error not exceeding \(0.5\%\).

In addition, from Fig. 1 we have

\[ \sin\alpha_1=\frac{H}{e-r} \tag{5} \]

and consequently,

\[ \cos\alpha_1=\sqrt{\,1-\left(\frac{H}{e-r}\right)^2\,}\cong 1-\frac{H^2}{2(e-r)^2} \tag{6} \]

Using expressions (4a), (5), and (6), and putting \(l\cong l-r\) in the terms that are small in magnitude, after the corresponding transformations we find

\[ x=r\left\{1+\frac{H^2}{2e^2} -\left[1+\frac{H^2}{2e^2}\right]\cos\alpha -\frac{H^3}{2e^3}\sin\alpha\right\} \tag{7} \]

The last term here is very small; therefore, with an error of no more than \(1\%\), it may be neglected; in that case

\[ x=r\left[1+\frac{H^2}{2e^2}\right][1-\cos\alpha] \tag{7a} \]

Finally, with the aid of expression (2-a), we obtain definitively

\[ \boxed{x=\frac{s}{2}[1-\cos\alpha]} \tag{7b} \]

This formula, despite being approximate (it gives an error of \(2\text{–}3\%\)), is nevertheless sufficiently accurate for purposes of investigation and practice, and in addition is very simple and convenient both for analytical and for graphical investigation of the operation of the cutting apparatus.

By applying formula (7b), we, as it were, replace the actual connecting-rod-and-crank mechanism by a fictitious one, with crank radius \(\dfrac{s}{2}\) and an infinitely long connecting rod.

Differentiating formula (7b) with respect to time and bearing in mind that

\[ \alpha=\omega t \tag{8} \]

where the angular velocity of the crank is

\[ \omega=\frac{\pi n}{30}, \tag{9} \]

and \(t\) is time, we obtain an expression for the relative velocity of the knife

\[ \boxed{U=\frac{s\omega}{2}\sin\alpha} \tag{10} \]

From this expression it is evident that when \(\alpha=0\) and \(\alpha=\pi\), i.e. in the extreme positions, the velocity of any point is equal to zero, while when \(\alpha=\frac{\pi}{2}\) and \(\frac{3\pi}{2}\), i.e. in the middle position, the velocity reaches a maximum

\[ U_{\max}=\frac{s\omega}{2}=\frac{\pi s n}{60}. \tag{11} \]

Under practical conditions the velocity of the knife is usually characterized by the so-called mean velocity, i.e. such a velocity at which the knife, moving uniformly, would in the time corresponding to a half-revolution of the crank pass the same path \(s\) as it does in reality. Obviously, the magnitude of the mean velocity is determined by the expression

\[ U_{\mathrm{cp}}=\frac{2sn}{60}=\frac{sn}{30}=\frac{2U_{\max}}{\pi}. \tag{12} \]

Transport motion.

In addition to its motion relative to the finger bar, the knife, together with the machine, moves over the field. This motion, for small sections of the path, may be considered, without any error, rectilinear and uniform, and is characterized by the so-called feed, i.e. by the displacement through which the machine moves forward during the time corresponding to one stroke of the knife. The magnitude of the feed will obviously be equal to

\[ h=\frac{v\cdot 60}{n\cdot 2}=\frac{30\cdot v}{n}; \tag{13} \]

where \(v\) is the speed of the machine in m/sec,
\(n\) is the number of revolutions of the crankshaft.

From formula (12) we have

\[ \frac{30}{n}=\frac{s}{U_{\mathrm{cp}}}=\frac{\pi}{2}\cdot\frac{s}{U_{\max}}. \]

Therefore formula (13) for determining the feed may be written in the form

\[ \boxed{h=s\frac{v}{U_{\mathrm{cp}}}=\frac{\pi s}{2}\cdot\frac{v}{U_{\max}}} \tag{13a} \]

If the feed of the machine is known, then the equation of the transport motion of the knife, as a function of the angle of rotation of the crank, may be written in the form

\[ Y=\frac{h}{\pi}\alpha \tag{14} \]

Here the initial position is taken to be the position which the point under consideration occupied when the knife was in its extreme right-hand position.

In fact, if a rotation of the crank through \(\pi\) radians corresponds to a displacement of the machine by \(h\), then a rotation by one radian will correspond to a displacement \(\pi\) times smaller, and a rotation by \(\alpha\) radians to a displacement \(\alpha\) times greater.

Absolute motion.

The absolute motion of any point of the knife, as the sum of the relative and translational motions, will be determined, in parametric form, by two equations (7b) and (14).

If we are interested in the trajectory of motion of some point, then the equation of this trajectory can be obtained by eliminating the angle of rotation of the crank \(\alpha\) from both equations.

In the case where several points of the knife are investigated simultaneously and, moreover, in the same coordinates, the equations of motion of these points will differ from one another by constant terms.

Fig. 2.

Thus, for example, if in Fig. 2 we denote by:

\(a\)—the width of the lower base of the segment,
\(b\)—the width of the upper base of the segment,
\(c\)—the working height of the segment,
\(m\)—the distance between the lower bases of the segments, then:

For point \(A\).

Equations of motion:

\[ \left. \begin{aligned} X &= \frac{s}{2}[1-\cos \alpha] + \frac{a}{2} \\ Y &= \frac{h}{\pi}\alpha \end{aligned} \right\} \tag{15a} \]

Equation of the trajectory:

\[ Y_a=\frac{h}{\pi}\arccos\left[1-\frac{2x-a}{s}\right]. \tag{16a} \]

For point \(B\).

Equations of motion

\[ \left. \begin{aligned} X &= \frac{s}{2}[1-\cos \alpha]+\frac{b}{2},\\ Y &= \frac{h}{\pi}\alpha+c \end{aligned} \right\} \tag{15b} \]

Equation of the trajectory

\[ Y_b=\frac{h}{\pi}\arccos\left[1-\frac{2x-b}{s}\right]+c \tag{16b} \]

For point \(C\).

Equations of motion

\[ \left. \begin{aligned} X &= \frac{s}{2}[1-\cos \alpha]-\frac{b}{2},\\ Y &= \frac{h}{\pi}\alpha+c \end{aligned} \right\} \tag{15c} \]

Equation of the trajectory

\[ Y_c=\frac{h}{\pi}\arccos\left[1-\frac{2x+b}{s}\right]+c \tag{16c} \]

For point \(D\).

Equations of motion

\[ \left. \begin{aligned} X &= \frac{s}{2}[1-\cos \alpha]-\frac{a}{2},\\ Y &= \frac{h}{\pi}\alpha \end{aligned} \right\} \tag{15d} \]

Equation of the trajectory

\[ Y_d=\frac{h}{\pi}\arccos\left[1-\frac{2x+a}{s}\right] \tag{16d} \]

For point \(E\).

Equations of motion

\[ \left. \begin{aligned} X &= \frac{s}{2}[1-\cos \alpha]-\frac{a}{2}-m,\\ Y &= \frac{h}{\pi}\alpha \end{aligned} \right\} \tag{15e} \]

Equation of the trajectory

\[ Y_e=\frac{h}{\pi}\arccos\left[1-\frac{2x+a+2m}{s}\right] \tag{16e} \]

The analytical investigation of the motion of the knife, generally speaking, may be used to resolve various questions connected with the operation of the cutting apparatus, but it has a major shortcoming, namely, a lack of visual clarity.

From this point of view it is more desirable to use a graphical representation of the trajectory of motion of some point of the knife. The method for constructing the trajectory is quite simple, follows entirely from the meaning of formulas \((b)\) and \((14)\), and reduces to the construction of a sinusoid.

Diagram of the motion of the segments and its analysis.

The cutting process.

Despite the fact that the design of the cutting apparatus was established quite a long time ago, there is still no single point of view on the cutting process. What seems indisputable is only that, at low speeds of knife motion, the stems are bent by the segment until the finger plate prevents this; the stem clamped between two blades is, as it were, sheared off by the segment advancing onto the finger plate; at high speeds of knife motion, the elasticity of the stem and the inertia of its mass increase so much that they prove sufficient for the stem to be cut without support from the finger plate; the magnitude of the critical speed, after which unsupported cutting becomes possible, will obviously depend on the kind of plant and on the state of its maturity, moisture content, etc.

Disagreements among specialists in the field of agricultural machines are mainly grouped around the question of the specific value of this critical speed. Some say that modern harvesting machines generally operate at speeds above the critical speed and, consequently, the cutting process occurs mainly according to the principle of unsupported cutting. Others say, on the contrary, that existing machines operate at speeds below the critical speed and that, consequently, cutting occurs according to the principle of scissors. A third group believes that both occur.

On the basis of the data of the experimental investigation by engineers A. N. Karpenko and E. A. Glukhikh, and on the facts of everyday practice with harvesting machines, the second point of view should be considered more correct; it holds that the essence of the operation of the cutting apparatus consists in the grasping of stems by the segments, bringing them to the finger plates, and, finally, cutting them off by the segment advancing onto the finger plate.

Diagram of the motion of the segments.

In accordance with the point of view set forth above, it must be assumed that all stems are brought up to the so-called cutting lines (Fig. 3) and, being clamped there between the blades of the finger plate and of the segment, are sheared off.¹ To find each such cutting line, it is sufficient to find any two of its points; for example, the point $d$ of the lower base of the working part of the segment will approach the point $d'_1$ of the finger plate only when the latter has moved into the position $d_1$; similarly, the point $c$ of the segment will approach the point $c'_1$ of the finger plate when the latter has moved into the position $c_1$. Obviously, the point $d_1$ will be the beginning of the cutting line, and the point $c_1$ its end. By connecting these points with a straight line, we obtain the cutting line for the stroke from left to right.

Considering the diagram obtained, we come to the conclusion that: on the section $Md c_1 L$ the stems are bent by the finger, from the axis of its symmetry, to the left and are cut on the blade of the finger plate of the same finger by the blade of the segment advancing onto it; on the section $RCDEFK$ the stems are bent by the finger, from the axis of its symmetry, to the right, and are then picked up by the segment emerging from the slot of this finger, and together with the stems from the section $d_1NRKFc_1$ are brought to the opposite finger, where they are re-

are cut; in the section \(ABNd_1M\) the stalks are bent further to the left, and then by the segment emerging from the slot of this finger during its motion from right to left; but since the segment stops before reaching the opposing finger, the stalks are picked up by the neighboring left segment during the motion of the latter from right to left and are brought to point \(d_1\) of the cutting line, where they are cut in a more or less compact bundle.

Fig. 3.

In the section \(dNR\) and \(QFSP\) the segments pass twice. The stems on them will be cut by the segment that passes over this section first.

Bending of stems

It follows from the above that all stems are cut in a bent state. Therefore the height of the stubble remaining in the field is never the same. This is especially noticeable after mowers have worked on wet grass. In this connection, the question of the bending of stems is not of particular fundamental importance, as the first investigators of the cutting apparatus thought (Wüst, Nachtweh, and others), but must be considered mainly from the economic point of view, since an increase in bending reduces the mass of mown hay when mowers operate and increases the danger of loss of ears when grain-harvesting machines operate.

The stems bent from the section \(ABNdM\) are in a somewhat special position, since they are gathered into a bunch and cut almost simultaneously. If these areas are large, a considerable increase in the cutting resistance, which has the character of an impact, is inevitable. If they are small, it must be assumed that their presence does no particular harm to the cutting apparatus. However, in the absence of corresponding experiments it is not possible to establish any limit for the permissible size of these areas.

Areas with double passage

On the diagram of segment motion there are two kinds of areas over which the active blades of the segments pass twice. Some areas with double passage arise because the region traversed by one blade of the segment is superposed on the region traversed by the other blade of the same segment.

These sections (\(dNR\) and \(d'N'R'\) in Fig. 3) have the form of triangles with curved sides, and when they are large they may be harmful in the sense that stems cut during the stroke of the segment in one direction may come under the blade of the segment during the return stroke and be cut again on another finger. The small pieces obtained in this secondary cut may contribute to clogging of the cutting apparatus, not to mention the wasteful expenditure of energy on the secondary cut. The danger of a secondary cut exists, generally speaking, also for stems growing adjacent to triangular areas, but since it is difficult to take this danger into account, in what follows we shall speak only of preventing the danger of a secondary cut for stems growing on triangular areas.

Obviously, a secondary cut can be avoided only in the case when these areas are located only in one half of the space between the fingers (for the section \(d'N'R'\), in the right half). In this case the stems cut during the stroke in one direction will slip out from under the segment during the return stroke before the blade of the segment reaches the finger plate.

Such a position can occur only when, in Fig. 2, the coordinates of the point \(z\) of intersection of the trajectories of points \(B\) and \(D\) of the segment satisfy the conditions

\[ Y_z = Y_b = Y_d \]

\[ X_z = X_b = X_d \geq \frac{t_1}{2} \]

where \(t_1\) is the distance between the fingers.

Combining these conditions with expressions (16 b) and (16 d), we can write for the limiting case

\[ X_b = X_d = \frac{t_1}{2} \]

\[ \frac{h}{\pi}\arccos\left[1-\frac{t_1+a}{s}\right] = \frac{h}{\pi}\arccos\left[1-\frac{t_1-b}{s}\right]+c \]

Whence

\[ h \ge \frac{\pi c}{ \arccos\left[1-\frac{t_1+a}{s}\right] - \arccos\left[1-\frac{t_1-b}{s}\right] } \tag{17} \]

The danger of a secondary cut on triangular-shaped areas occurs mainly in mowing machines, for which \(t_1=s=a_1\); therefore, instead of (17) we obtain the condition preventing the danger of a secondary cut for mowing machines in the form

\[ \boxed{ h \ge \frac{\pi c}{ \arccos\left(-\frac{a}{a_1}\right) - \arccos\left(\frac{b}{a_1}\right) } } \tag{17a} \]

where \(a_1\) is the width of the segment.

In this formula the first term of the denominator must be taken in the III quadrant, and the second in the I or II quadrant.

In binders and combines the diagram of the motion of the segments has a somewhat different form, and the areas with double travel (of triangular shape) are either entirely absent or else insignificant; therefore formula (17) is not applicable to them.

Other regions with double travel arise because the right or left blades of neighboring segments pass over one and the same place. These areas (\(QFSP\) in Fig. 3) in mowing machines need not be considered especially harmful, since the stems cut by the segment going ahead are only bent aside by the second segment, and therefore one can speak only of the expenditure of energy for bending, which in any case is insignificant.

The matter is somewhat different in combines. Here these areas prove to be elongated and extend over a distance exceeding the magnitude of the finger pitch. Therefore part of the blade of the segment passes over already cut stems without performing any work. Obviously, it must be considered desirable to eliminate these areas and to enable the blade of the segment to work with its full length. For this, in turn, it is necessary that the trajectory of point \(E\) of the segment (Fig. 2) should not intersect the trajectory of point \(B\) of the segment, but only touch the latter.

The point of intersection of these trajectories will be determined by the conditions:

\[ X_e = X_b = X_z \]

\[ Y_e = Y_b = Y_z \]

Therefore, bearing in mind formulas (15 b) and (15 e), we can write

\[ \frac{s}{2}\left[1-\cos\alpha_b\right]+\frac{b}{2} = \frac{s}{2}\left[1-\cos\alpha_e\right]-\frac{a}{2}-m \]

Whence

\[ \frac{b+a+2m}{s}=\cos \alpha_b-\cos \alpha_e \]

but

\[ \cos \alpha_b-\cos \alpha_e =\cos \frac{\pi(Y_z-c)}{h}-\cos \frac{\pi Y_z}{h} =2\sin \frac{\pi(2Y_z-c)}{2h}\sin \frac{\pi c}{2h}. \]

Therefore

\[ \frac{b+a+2m}{s}=2\sin \frac{\pi(2Y_z-c)}{2h}\sin \frac{\pi c}{2h}. \]

Whence

\[ Y_z=\frac{h}{\pi}\arcsin\left[\frac{b+a+2m}{2s\sin \frac{\pi c}{2h}}\right]+\frac{c}{2}. \tag{18} \]

If the trajectory of point \(E\) is tangent to point \(B\), then \(Y\) must have only one value, and this can occur only in the case when

\[ \frac{a+b+2m}{2s\sin \frac{\pi c}{2h}}=1. \]

Whence

\[ h=\frac{\pi c}{2\arcsin\left(\frac{a+b+2m}{2s}\right)}; \tag{19} \]

but for combines \(m=0\) and \(s=2t_1=a\), therefore

\[ \boxed{ h=\frac{\pi c}{2\arcsin \frac{1}{2}\left[1+\frac{b}{a}\right]} } \tag{19a} \]

3. Angles of inclination of the segment blade and the finger plate.

From the diagram of the motion of the segments (Fig. 3) it is evident that it is desirable for us to have the angles of inclination of the segment blade \(\alpha\) and of the finger plate \(\beta\) as large as possible, since by this we increase the duration of cutting and, consequently, reduce the resistance.

However, there is a limit to increasing these angles. With respect to the angle \(\beta\), this limit is established practically by the circumstance that, at a greater value of the angle \(\beta\), the finger plate will press on the stems and deflect them in the direction of motion of the machine. Therefore \(\beta \leq 7^\circ\) is usually used, and in combines even \(\beta=0\). In this connection, an increase in the duration of cutting can be obtained mainly by increasing the angle of inclination of the segment blade \(\alpha\).

The task of the segment in the operation of the cutting apparatus consists in bringing the stems up to the finger plates and then cutting them; therefore, in increasing the angle \(\alpha\), we must see to it that the segment can perform the tasks assigned to it.

Indeed, one can imagine so large an angle \(\alpha\) at which the stems will slip forward out from under the blade of the segment either at the moment of cutting them, or even while they are being brought up to the finger-hasp?

plate. Therefore it is obvious that, in increasing the angle \(\alpha\), we must not pass beyond the limit after which the stems will slip out from under the blade of the segment.

In order to determine the limit for the angle \(\alpha\), let us consider the motion of the segment relative to the stems. Relative to the finger bar, the segment moves with a velocity variable in magnitude,

\[ U = U_{\max}\sin\omega t, \]

directed at an angle \(90-\alpha\) to the blade of the segment. In addition, together with the machine, the segment moves forward with velocity \(v\), directed at an angle \(\alpha\) to the blade of the segment (Fig. 4).

The absolute velocity of the segment will obviously be equal to their geometric sum \(w\) and may be resolved into two directions: parallel to the blade and perpendicular to it.

The first component:

\[ w' = U_{\max}\sin\alpha\cdot\sin\omega t - v\cos\alpha \tag{20} \]

gives the velocity with which the segment, as it were, slides relative to the stems.

The second component:

\[ w'' = U_{\max}\cos\alpha\cdot\sin\omega t + v\sin\alpha \tag{21} \]

is the velocity with which the segment presses upon the stems.

Stems can slip out from under the blade of the segment only in the direction toward the upper base, and moreover only in the case when the sliding velocity is directed toward the lower base of the segment. Therefore, if there were no friction of the segment against the stem, it would be necessary that \(w' \le 0\). But owing to the fact that the segment will also hold the stems by the force of friction, \(w'\) may be allowed to be somewhat greater than zero, namely such that the absolute velocity \(w\) deviates from the perpendicular to the blade of the segment by an angle not exceeding the angle of friction \(\varphi_{\text{seg}}\). In other words, the stems will not slip out from under the blade of the segment if

\[ \frac{w'}{w''} \le \operatorname{tg}\varphi_{\text{seg}}. \]

Fig. 4.

Substituting here expressions (20) and (21), we obtain

\[ \operatorname{tg}\alpha \le \frac{\dfrac{v}{U_{\max}}+\operatorname{tg}\varphi_{\text{seg}}\cdot\sin\omega t} {\sin\omega t-\dfrac{v}{U_{\max}}\operatorname{tg}\varphi_{\text{seg}}} \tag{22} \]

The least value of the right-hand side will be at \(\omega t=\pi/2\); therefore, denoting

\[ \frac{v}{U_{\max}} = k, \tag{23} \]

we find the limiting condition for the angle \(\alpha\) in the form

\[ \operatorname{tg}\alpha \le \frac{k+\operatorname{tg}\varphi_{\text{seg}}} {1-k\,\operatorname{tg}\varphi_{\text{seg}}} \tag{22a} \]

Thus, the magnitude of the angle of inclination of the segment blade that guarantees the stems against slipping will be greater the larger the ratio of the speed of the machine to the speed of the knife, and the greater the angle of friction of the segment against the stem.

In addition, the stems also must not slip out from under the blades pressing them at the moment of cutting. In this case (Fig. 5), the stem will press on the segment with a certain force \(P\), and on the other side the ledger plate with a force \(Q\). Under the action of these forces, the corresponding friction forces will arise

Fig. 5.

\[ F_{\text{seg}} = f_{\text{seg}}\cdot P \qquad \text{and} \qquad F_{\text{pl}} = f_{\text{pl}} Q \]

The stem cannot move in the direction of motion of the knife; therefore

\[ P\cos \alpha - Q\cos \beta + F_{\text{seg}}\sin \alpha - F_{\text{pl}}\sin \beta = 0 \]

In addition, in order to avoid pushing the stem out, it is necessary that

\[ P\sin \alpha + Q\sin \beta \leq F_{\text{seg}}\cos \alpha + F_{\text{pl}}\cos \beta \]

Solving the last two expressions jointly, after the corresponding transformations, we find

\[ \boxed{\alpha + \beta \leq \varphi_{\text{seg}} + \varphi_{\text{pl}}} \tag{24} \]

Thus, in order that the stems at the moment of cutting not slip out from under the blades pressing them, it is necessary that the sum of the angles of inclination of the segment and of the ledger plate be less than the sum of the angles of friction of the stem against the blade.

Width of the Segment.

Tractor Mower.

For mowers we earlier derived a condition guaranteeing against secondary cutting,

\[ h > \frac{\pi c}{\arccos\left(-\frac{a}{a_1}\right) - \arccos \frac{b}{a_1}}; \tag{17a} \]

In addition, we have the formula

\[ h=\frac{\pi s}{2}\frac{v}{U_{\max}}=\frac{\pi a_1 k}{2} \tag{13a} \]

From Fig. 6 we have the relation between the dimensions of the segments

\[ a=a_1-2\Delta\tg\alpha \tag{25} \]

\[ c=\frac{a-b}{2\tg\alpha} =\frac{a_1-b-2\Delta\tg\alpha}{a_1 k\tg\alpha} \tag{26} \]

Fig. 6.

Substituting (25) and (26) into (17a) and equating the result obtained to (13a), we obtain the condition for choosing \(a_1=t_1=s\)

\[ \arccos\left(\frac{2\Delta\tg\alpha}{a_1}-1\right) -\arccos\frac{b}{a_1} \ge \frac{[a_1-(b+2\Delta\tg\alpha)]}{a_1 k\tg\alpha}. \tag{27} \]

For example, if we take

\(v=1.33\ \text{m/sec}\), as the operating speed of the STZ tractor,
\(U_{\text{av}}=1.70\ \text{m/sec}\), as the lowest speed in existing designs of mowing machines (according to Weis, it is sufficient that \(v=1.5\)), then

\[ U_{\max}=\frac{\pi}{2}\cdot 1.7=2.66\ \text{m/sec};\qquad k=\frac{1.33}{2.66}=0.5 \]

\(\alpha=30^\circ\)—according to existing designs,
\(\Delta=8\ \text{mm}\)—as the smallest dimension permitting the formation of supporting whiskers,
\(b=15\ \text{mm}\)—as the smallest dimension permitting the possibility of placing the blade bevels, then by selection (or by constructing graph 7) one can find from formula (27)

\[ a_1=t_1=s=73\ \text{mm}. \]

The result, as we see, is very close to that used in practice.

By means of expansions in series it would be possible to reduce formula (27) to algebraic form and, solving the latter, obtain the formula

\[ a_1=t_1=s= \frac{b(1+k\tg\alpha)+2\Delta\tg\alpha(1+2.46\,k\tg\alpha)} {1-1.82\,k\tg\alpha} \tag{27a} \]

which is simpler to solve, but gives an error of up to 10%, and therefore it is better to solve equation (27) by selection.

The “Kommunar” Combine

Earlier a condition was derived that guarantees the absence of areas of double travel:

\[ h=\frac{\pi c}{2\arcsin \frac{1}{2}\left(1+\frac{b}{a}\right)} \qquad \cdots \cdots \cdots \cdots \cdots (19a) \]

In addition, we have

\[ h=\frac{\pi s}{2}\cdot \frac{v}{U_{\max}} =\frac{\pi a k}{2} \qquad \cdots \cdots \cdots \cdots \cdots (13a) \]

Figure labels: solution of an equation of the form

\[ \arccos\left(\frac{2a\tan\alpha}{a_1}-1\right)-\arccos\frac{b}{a_1} = \frac{a_1-(b+2\ldots)}{a_1 k\tan\alpha} \]

for \(\Delta=8\), \(\alpha=30^\circ\), \(k=0.5\). The graph also bears the curve labels \(b=14\), \(b=15\), \(b=20\), and the axis label \(a_1\).

Fig. 7.

Finally, formula (26) for the combine at \(\Delta=0\) will have the form

\[ c=\frac{a-b}{2\tan\alpha} \qquad \cdots \cdots \cdots \cdots \cdots (26a) \]

Substituting (26a) into (19a) and equating the result obtained to (13a), we obtain the condition for choosing \(a=s=2t\):

\[ \frac{1}{2}\left(1-\frac{b}{a}\right) = k\tan\alpha\cdot \arcsin \frac{1}{2}\left(1+\frac{b}{a}\right) \qquad \cdots \cdots \cdots (28) \]

For example, if we take:

\(v = 1.27\ \text{m/sec.}\), as the operating speed of the STZ tractor when working with a combine (see the Agricultural Encyclopedia, “Combine”), \(U_{\mathrm{av}} = 1.11\ \text{m/sec.}\) at \(n = 327\),

\[ U_{\max} = 1.11 \frac{\pi}{2} = 1.75\ \text{m/sec.}, \qquad k = \frac{1.27}{1.75} = 0.73, \]

\(b = 5\ \text{mm}\), as the smallest dimension ensuring the strength of the segment tip, adopted for combines, then, solving equation (28) by selection or graphically, we find

\[ a = s = 2t_1 = 110\ \text{mm}. \]

The result again proves to be close to the dimension existing in practice.

In the particular case where \(b = 0\), from formula (28) we have

\[ k \tg \alpha = \frac{1}{2 \arcsin 0.5} = 0.95; \qquad k = \frac{0.95}{\tg \alpha} \tag{29} \]

i.e., irrespective of the magnitude of \(a\), the areas of double travel will be absent if condition (29) alone is satisfied.

References

1. Prof. V. P. Goryachkin. Theory of Reaping Machines. 1909.
2. Prof. L. P. Kramarenko. The Cutting Apparatus of Reaping Machines and Its Theory.
3. Prof. L. P. Kramarenko. Articles in the journal Agricultural Machine. 1931, No. 7.
4. E. A. Pukhikh. On the Question of Increasing the Forward Speed of Reaping Machines. Journal Mechanization of Socialist Agriculture. 1934, No. 9.
5. A. N. Karpenko. Articles in the journal Agricultural Machine. 1931, Nos. 9, 11–12.

1. See the article by Prof. A. V. Verkhovsky, “The working process of a mowing machine.” Izv. T. I. I., 1935. ↩