On the Problem of Balancing the Connecting Rod-Crank Mechanism of Harvesting Machines
Abstract Generated abstract
This paper analyzes the balancing of connecting rod and crank mechanisms in harvesting machines, where practical treatments of connecting rod mass distribution and counterweighting were inconsistent. It derives approximate kinematic and dynamic formulas for the crank, connecting rod, and knife, then uses them to express inertia forces and moments for the mechanism and to compare several mower, binder, header, and combine types. The study argues that, for inertia force calculations, the connecting rod mass should be divided between the crank pin and knife head in inverse proportion to the distances from its center of gravity, rather than by a fixed fraction. It further concludes that complete balancing of both inertia forces and their moments by counterweights is not practically attainable, while balancing rotating masses alone is feasible and generally preferable.
Full Text
Eng. G. D. TERSKOV.
ON THE PROBLEM OF BALANCING THE CONNECTING ROD–CRANK MECHANISM OF HARVESTING MACHINES.
Introduction.
Despite the fact that the theory of the connecting rod–crank mechanism in general has now been developed in sufficient detail, its application to agricultural machines still suffers from a certain uncertainty.
As evidence of this uncertainty, for example, one may cite the fact that, in determining inertia forces, some authors distribute the mass of the connecting rod as follows: to the knife head—\(2/3\), and to the crank pin—\(1/3\) of the mass; others, however, distribute it taking into account the position of the center of gravity.
In addition, one sometimes encounters the assertion that, with the aid of two counterweights, it is possible, theoretically, to balance the connecting rod–crank mechanism completely.
The purpose of the present work is to bring clarity to the application of the general conclusions of mechanics to the connecting rod–crank mechanism of harvesting agricultural machines, and, in addition, to determine concretely the magnitudes reached by the inertia forces in these machines.
I. KINEMATICS OF THE CONNECTING ROD–CRANK MECHANISM.
A. Connecting rod.
Let us consider the connecting rod–crank mechanism at some moment of its motion (Fig. No. 1), when the crank makes an angle with the horizontal line
\[ \varphi = \omega t, \ldots\ldots\ldots\ldots\ldots (1) \]
where \(\omega\) is the angular velocity of the crank, which, generally speaking, varies within certain limits; but in what follows we shall consider it constant. In this case, if \(n\) is the number of revolutions of the crankshaft, then the angular velocity is determined by the formula
\[ \omega = \frac{\pi n}{30}\ldots\ldots\ldots\ldots\ldots (2) \]
At the instant under consideration the connecting rod makes an angle \(\psi\) with the horizontal. The dependence of this angle on the angle \(\varphi\) is determined from the following considerations:
on the one hand, from triangle \(ABA_1\),
\[ AA_1 = l \operatorname{Sn}\psi, \]
on the other hand, from the quadrilateral \(AO_1OA_1\),
\[ AA_1 = a + r \operatorname{Sn}\varphi. \]
Therefore
\[ l \operatorname{Sn}\psi = a + r \operatorname{Sn}\varphi \]
or
\[ \operatorname{Sn}\psi=\frac{a+r\operatorname{Sn}\varphi}{l} \ldots \ldots \ldots \ldots \ldots (3) \]
Introducing the ratios
\[ k=\frac{a}{r}; \qquad \lambda=\frac{r}{e} \ldots \ldots \ldots \ldots \ldots (4) \]
we obtain
\[ \operatorname{Sn}\psi=\lambda(k+\operatorname{Sn}\varphi) \ldots \ldots \ldots \ldots \ldots (3a) \]
Let us take on the connecting rod \(AB\) some arbitrary point \(N\), lying at a distance \(u\) from the crank pin \(A\). The coordinates of this arbitrary point will be determined by the relations
\[ \begin{aligned} x &= u \operatorname{Cos}\psi-r\operatorname{Cos}\varphi\\ z &= (l-u)\operatorname{Sn}\psi \end{aligned} \]
or, eliminating from here the angle \(\psi\) with the aid of (3a), we obtain
\[ \begin{aligned} x &= u\sqrt{1-\lambda^2(k+\operatorname{Sn}\varphi)^2}-r\operatorname{Cos}\varphi\\ z &= (l-u)\lambda(k+\operatorname{Sn}\varphi) \end{aligned} \ldots \ldots \ldots (5) \]
Since the second term under the radical is very small in comparison with unity—namely, it does not exceed:
for mowers—\(0.13\)
and for binders—\(0.20\),
then, with sufficient accuracy for practice (with an error of less than \(1\%\)), we may apply the approximate formula for extracting the square root:
\[ \sqrt{1-\delta}\simeq 1-\frac{1}{2}\delta. \]
Then
\[ \sqrt{1-\lambda^2(k+\operatorname{Sn}\varphi)^2} = 1-\frac{\lambda^2}{2}(k+\operatorname{Sn}\varphi)^2 = 1-\frac{\lambda^2k^2}{2} -\lambda^2 k\operatorname{Sn}\varphi -\frac{\lambda^2}{2}\operatorname{Sn}^2\varphi. \]
And since
\[ \operatorname{Sn}^2\varphi=\frac{1}{2}-\frac{1}{2}\operatorname{Cos}2\varphi \]
then
\[ \sqrt{1-\lambda^2(k+\operatorname{Sn}\varphi)^2} = 1-\frac{\lambda^2k^2}{2} -\frac{\lambda^2}{4} -\lambda^2 k\operatorname{Sn}\varphi +\frac{\lambda^2}{4}\operatorname{Cos}2\varphi. \tag{6} \]
Then the equations of motion of any point of the connecting rod will have the form:
\[ \begin{aligned} x&=u\left(1-\frac{\lambda^{2}k^{2}}{2}-\frac{\lambda^{2}}{4}-\lambda^{2}k\,\operatorname{Sn}\varphi+\frac{\lambda^{2}}{4}\operatorname{Cos}2\varphi\right)-r\operatorname{Cos}\varphi,\\ z&=(l-u)\lambda(k+\operatorname{Sn}\varphi) \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (5a) \end{aligned} \]
Taking expression (1) into account and differentiating formulas (5a) with respect to time, we find the equations of the velocity of any point of the connecting rod.
\[ \begin{aligned} \frac{dx}{dt}&=-\omega\left[u\left(\lambda^{2}k\operatorname{Cos}\varphi+\frac{\lambda^{2}}{2}\operatorname{Sn}2\varphi\right)-r\operatorname{Sn}\varphi\right],\\ \frac{dz}{dt}&=\omega(l-u)\lambda\operatorname{Cos}\varphi \ldots \ldots \ldots \ldots \ldots \ldots \ldots (7) \end{aligned} \]
In turn, differentiating (7) with respect to time, we find the equations of acceleration of any point of the connecting rod:
\[ \begin{aligned} \frac{d^{2}x}{dt^{2}}&=\omega^{2}\left[u\left(\lambda^{2}k\operatorname{Sn}\varphi-\lambda^{2}\operatorname{Cos}2\varphi\right)+r\operatorname{Cos}\varphi\right],\\ \frac{d^{2}z}{dt^{2}}&=-\omega^{2}(l-u)\lambda\operatorname{Sn}\varphi \ldots \ldots \ldots \ldots \ldots \ldots \ldots (8) \end{aligned} \]
In particular, if the center of gravity of the connecting rod is located on the line \(AB\) (which we shall henceforth call the axis of the connecting rod) at a distance \(s\) from the crank pin, then the law of motion of the center of gravity of the connecting rod is determined by the equations:
\[ \begin{aligned} x_{\sh}&=s\left(1-\frac{\lambda^{2}k^{2}}{2}-\frac{\lambda^{2}}{4}-\lambda^{2}k\operatorname{Sn}\varphi+\frac{\lambda^{2}}{4}\operatorname{Cos}2\varphi\right)-r\operatorname{Cos}\varphi,\\ z_{\sh}&=(l-s)\lambda(k+\operatorname{Sn}\varphi) \ldots \ldots \ldots \ldots \ldots \ldots \ldots (5в) \end{aligned} \]
\[ \begin{aligned} v_{\sh}^{x}&=\omega\left[r\operatorname{Sn}\varphi-s\left(\lambda^{2}k\operatorname{Cos}\varphi+\frac{\lambda^{2}}{2}\operatorname{Sn}2\varphi\right)\right],\\ v_{\sh}^{z}&=\omega(l-s)\lambda\operatorname{Cos}\varphi \ldots \ldots \ldots \ldots \ldots \ldots \ldots (7a) \end{aligned} \]
\[ \begin{aligned} j_{\sh}^{x}&=\omega^{2}\left[r\operatorname{Cos}\varphi+s\left(\lambda^{2}k\operatorname{Sn}\varphi-\lambda^{2}\operatorname{Cos}2\varphi\right)\right],\\ j_{\sh}^{z}&=-\omega^{2}(l-s)\lambda\operatorname{Sn}\varphi \ldots \ldots \ldots \ldots \ldots \ldots \ldots (8a) \end{aligned} \]
Formulas (5в), (7a), (8a) could completely characterize the motion of the connecting rod only in the case where the connecting rod moved translationally. But since the connecting rod has a complex motion, which, according to the laws of theoretical mechanics, can be decomposed into translational and rotational motion, then in addition to the above formulas it is necessary to give the law of the rotational motion of the connecting rod.
Obviously (from formula 3a) the angle \(\psi\) varies according to the law:
\[ \psi=\arcsn\,\lambda(k+\operatorname{Sn}\varphi) \ldots \ldots \ldots \ldots (9) \]
The angular velocity is found by differentiating formula (9)
\[ \frac{d\psi}{dt} = \frac{\lambda\omega\operatorname{Cos}\varphi} {\sqrt{1-\lambda^2(k+\operatorname{Sn}\varphi)^2}} . \]
Extracting the square root approximately, we obtain
\[ \frac{1}{\sqrt{1-\lambda^2(k+\operatorname{Sn}\varphi)^2}} = 1+\frac{\lambda^2}{2}(k+\operatorname{Sn}\varphi) = 1+\frac{\lambda^2 k^2}{2} + \]
\[ +\lambda^2 k\operatorname{Sn}\varphi + \frac{\lambda^2}{4} - \frac{\lambda^2}{4}\operatorname{Cos}2\varphi . \]
Then
\[ \frac{d\psi}{dt} = \lambda\omega\operatorname{Cos}\varphi \left[ 1+\frac{\lambda^2 k^2}{2} +\lambda^2 k\operatorname{Sn}\varphi +\frac{\lambda^2}{4} -\frac{\lambda^2}{4}\operatorname{Cos}2\varphi \right]. \tag{10} \]
Differentiating (10), we obtain the angular acceleration of the connecting rod
\[ \frac{d^2\psi}{dt^2} = \lambda\omega^2 \left[ \lambda^2 k\operatorname{Cos}2\varphi - \left( 1+\frac{\lambda^2 k^2}{2} -\lambda^2 +\frac{3}{2}\lambda^2\operatorname{Sn}^2\varphi \right) \operatorname{Sn}\varphi \right]. \tag{11} \]
B. Crank.
The crank, as already noted, has uniform rotary motion with angular velocity \(\omega\), and the equations characterizing the law of motion of the crank pin may be obtained from the corresponding equations for any point of the connecting rod by substituting \(u=0\) in them.
Then
\[ x=-r\operatorname{Cos}\varphi \]
\[ z=r(k+\operatorname{Sn}\varphi)=a+r\operatorname{Sn}\varphi \ldots \tag{12} \]
\[ \frac{dx}{dt}=\omega r\operatorname{Sn}\varphi \]
\[ \frac{dz}{dt}=\omega r\operatorname{Cos}\varphi \ldots \tag{13} \]
\[ \frac{d^2x}{d^2t}=\omega^2 r\operatorname{Cos}\varphi \]
\[ \frac{d^2z}{dt^2}=-\omega^2 r\operatorname{Sn}\varphi \ldots \tag{14} \]
If the center of gravity of the crank lies on the line \(O_1A\) (the crank axis) at a distance \(\varepsilon\) from the axis of rotation of the crank (toward the pin), then the equations characterizing the motion of the center of gravity of the crank may be obtained from (12, 13, 14) by mul-
multiplying them by the ratio \(\dfrac{\varepsilon}{r}\) (the distance \(a\) thereby remains unchanged).
\[ \begin{aligned} x_{kp} &= -\varepsilon \operatorname{Cos}\varphi,\\ z_{kp} &= a+\varepsilon \operatorname{Sn}\varphi \end{aligned} \qquad \dots \qquad (12a) \]
\[ \begin{aligned} v^{x}_{kp} &= \omega \varepsilon \operatorname{Sn}\varphi,\\ v^{z}_{kp} &= \omega \varepsilon \operatorname{Cos}\varphi \end{aligned} \qquad \dots \qquad (13a) \]
\[ \begin{aligned} j^{x}_{kp} &= \omega^{2}\varepsilon \operatorname{Cos}\varphi,\\ j^{z}_{kp} &= -\omega^{2}\varepsilon \operatorname{Sn}\varphi \end{aligned} \qquad \dots \qquad (14a) \]
C. Knife.
The knife performs rectilinear reciprocating motion, and therefore we obtain a complete representation of its motion by replacing, in the formulas for any point of the connecting rod, \(u\) by \(l\):
\[ x_{n}=r\left(\frac{1}{\lambda}-\frac{\lambda k^{2}}{2}-\frac{\lambda}{4}-\operatorname{Cos}\varphi-\lambda k\operatorname{Sn}\varphi+\frac{\lambda}{4}\operatorname{Cos}2\varphi\right) \]
\[ z_{n}=0 \qquad \dots \qquad (15) \]
\[ v^{x}_{n}=\omega r\left(\operatorname{Sn}\varphi-\lambda k\operatorname{Cos}\varphi-\frac{\lambda}{2}\operatorname{Sn}2\varphi\right) \]
\[ v^{z}_{n}=0 \qquad \dots \qquad (16) \]
\[ j^{x}_{n}=\omega^{2}r\left(\operatorname{Cos}\varphi+\lambda k\operatorname{Sn}\varphi-\lambda\operatorname{Cos}2\varphi\right) \]
\[ j^{z}_{n}=0 \qquad \dots \qquad (17) \]
II. INERTIA OF THE MASSES OF THE CRANK-CONNECTING-ROD MECHANISM.
As was already noted earlier, theoretical mechanics permits, for purposes of simplification, the complex motion of some figure to be regarded as if composed of two motions: 1) translational, in which the center of gravity moves according to the same law as in reality, and 2) rotational—about the center of gravity.
From the dynamic standpoint, translational motion is characterized by the fact that for it the mass of the body may be considered concentrated at the center of gravity; therefore, if \(m\) is the mass of a translationally moving body, then the projections of the inertia forces on the coordinate axes may be determined by the formulas
\[ I_{x}=-m\frac{d^{2}x_{c}}{dt^{2}} \]
\[ I_{z}=-m\frac{d^{2}z_{c}}{dt^{2}}, \qquad \dots \qquad (18) \]
and the moment of these forces with respect to the origin of coordinates, if counterclockwise is taken as positive, by the formula:
\[ M_{\mathrm{in.}}=I_{x}\cdot z-I_{z}\cdot x \qquad \dots \qquad (19) \]
Rotational motion, in turn, owing to its non-uniformity, will cause the appearance of an inertial couple, whose moment is determined by the relation:
\[ M_{\text{in. p.}}=-m\rho_0^2 \frac{d^2\psi}{dt^2}\ldots\ldots\ldots (20) \]
where \(\rho_0\) is the radius of inertia of the body relative to the center of gravity.
Using formulas 18, 19, 20, let us determine the inertia of the elements of the crank-and-connecting-rod mechanism.
A. Crank.
Inertia forces
\[ \begin{aligned} I_x^{\text{cr}}&=-m_{\text{cr}}\omega^2\varepsilon \cos\varphi\\ I_z^{\text{cr}}&=-m_{\text{cr}}\omega^2\varepsilon \sin\varphi \end{aligned} \ldots\ldots\ldots (21) \]
Moment of the inertia forces
\[ M_{\text{in.}}^{\text{cr}} = -m_{\text{cr}}\left[(a+\varepsilon)\sin\varphi\,\omega^2\varepsilon\cos\varphi -(-\varepsilon\cos\varphi)(-\omega^2\varepsilon\sin\varphi)\right] \]
After the appropriate transformations we obtain
\[ M_{\text{in.}}^{\text{cr}}=-m_{\text{cr}}a\omega^2\varepsilon\cos\varphi\ldots\ldots\ldots (22) \]
The moment of the inertial couple of the crank is equal to zero, since its rotation is uniform.
B. Knife.
Inertia forces
\[ \begin{aligned} I_x^{\text{k}}&=-m_{\text{k}}\omega^2 r(\cos\varphi+\lambda k\sin\varphi-\lambda\cos 2\varphi)\\ I_z^{\text{k}}&=0 \end{aligned} \ldots\ldots\ldots (23) \]
The moment of the inertia forces is equal to zero, since \(I_x^{\text{k}}\) passes through the origin of coordinates.
The moment of the inertial couple is also equal to zero, since the motion of the knife is translational.
C. Connecting rod.
Inertia forces
\[ I_x^{\text{r}} = -m_{\text{r}}\omega^2 r \left[ \cos\varphi+\frac{s}{l}(\lambda k\sin\varphi-\lambda\cos 2\varphi) \right] \]
\[ I_z^{\text{r}} = m_{\text{r}}\omega^2 r\frac{l-s}{l}\sin\varphi \ldots\ldots\ldots (24) \]
Moment of the inertia forces
\[ M_{\text{in}}^{\text{r}} = -m_{\text{r}}\frac{l-s}{l}r(k+\sin\varphi)\omega^2 r \left\{ \cos\varphi+\frac{s}{l}(\lambda k\sin\varphi-\lambda\cos 2\varphi) \right\} - \]
\[ -m_{\text{r}}\omega^2 r\frac{l-s}{l}\sin\varphi \left\{ -r\left[ \cos\varphi-\frac{s}{l} \left( \frac{1}{\lambda} -\frac{\lambda k^2}{2} -\frac{\lambda}{4} -\lambda k\sin\varphi +\frac{\lambda}{4}\cos 2\varphi \right) \right] \right\} \]
After the corresponding transformations we obtain:
\[ M^{\mathrm{sh}}_{\mathrm{in}} = - m_{\mathrm{sh}} \frac{l-s}{l}\,\omega^2 r^2 \left\{ k \Cos \varphi - \frac{s}{l} \left[ \lambda k \Cos 2\varphi - \left( \frac{1}{\lambda} + \frac{\lambda k^2}{2} - \lambda + \frac{3}{2}\lambda \Sn^2\varphi \right) \Sn \varphi \right] \right\} \ . \ . \ . \ . \ . \ (25) \]
Moment of the inertia couple
\[ M^{\mathrm{sh}}_{\mathrm{in.\,p.}} = - m_{\mathrm{sh}} \rho_0^2 \omega^2 \lambda^2 \left[ \lambda k \Cos 2\varphi - \left( \frac{1}{\lambda} + \frac{\lambda k^2}{2} - \lambda + \frac{3}{2}\lambda \Sn^2\varphi \right) \Sn \varphi \right] \ . \ . \ (26) \]
D. The entire mechanism.
Adding the inertia forces of all parts of the connecting-rod--crank mechanism, we obtain
\[ I_x = I_x^{\mathrm{cr}} + I_x^{\mathrm{sh}} + I_x^{\mathrm{k}} = -\omega^2 r \left[ \left( m_{\mathrm{cr}}\frac{\varepsilon}{r} + m_{\mathrm{sh}}\frac{l-s}{l} \right) \Cos \varphi + \left( m_{\mathrm{k}} + m_{\mathrm{sh}}\frac{s}{l} \right) \left( \Cos \varphi + \lambda k \Sn \varphi - \lambda \Cos 2\varphi \right) \right] \ . \ . \ . \ . \ . \ (27) \]
\[ I_z = I_z^{\mathrm{cr}} + I_z^{\mathrm{sh}} + I_z^{\mathrm{k}} = \omega^2 r \left( m_{\mathrm{cr}}\frac{\varepsilon}{r} + m_{\mathrm{sh}}\frac{l-s}{l} \right) \Sn \varphi \ . \ . \ (28) \]
\[ M_{\mathrm{in}} = M_{\mathrm{in}}^{\mathrm{cr}} + M_{\mathrm{in}}^{\mathrm{sh}} + M_{\mathrm{in.\,p.}}^{\mathrm{sh}} = -\omega^2 r \left\{ \left( m_{\mathrm{cr}}\frac{\varepsilon}{r} + m_{\mathrm{sh}}\frac{l-s}{l} \right) a\Cos \varphi - m_{\mathrm{sh}}\frac{ls-\rho_a^2}{l} \left[ \lambda^2 k \Cos 2\varphi - \left( 1 + \frac{\lambda^2 k^2}{2} - \lambda^2 + \frac{3}{2}\lambda^2 \Sn^2\varphi \right) \Sn \varphi \right] \right\} \ . \ . \ (29) \]
where \(\rho_a\) is the radius of inertia of the connecting rod relative to the crank pin.
Comparing the expressions obtained, which characterize the inertia of the masses of the entire connecting-rod--crank mechanism as a whole, with formulas (14), (17), and (11), the following conclusion can be drawn.
“When determining inertia forces, the entire connecting-rod--crank mechanism may be replaced by two masses:
a) the first mass is concentrated on the crank pin and rotates uniformly about the axis of the crankshaft; its magnitude is determined by the sum
\[ \left[ m^{\mathrm{cr}}\frac{\varepsilon}{r} + m_{\mathrm{sh}}\frac{l-s}{l} \right]; \]
b) the second mass is concentrated on the knife head and has rectilinear reciprocating motion; the magnitude of this mass is equal to
\[ \left[ m_{\mathrm{sh}}\frac{s}{l} + m_{\mathrm{k}} \right]. \]”
In other words: in determining the inertia forces of the connecting-rod--crank mechanism, the mass of the connecting rod may be considered distributed between the crank pin and the knife head inversely proportional to the distances of the center of gravity of the connecting rod from these points.
III. INERTIA OF THE MASS OF THE CRANK–CONNECTING-ROD MECHANISM OF THE MAIN TYPES OF HARVESTING MACHINES
Basic quantities characterizing the mechanisms of harvesting machines.
Table No. 1.
| Quantity | Tractor mower M. K. D. 7 | Horse mower G. L. 3 | Horse mower G. L. 3 | Horse-drawn reaper Kr. Aksai | Self-binder Kommunar | Combine (Windrower) Holt. 36 |
|---|---|---|---|---|---|---|
| Driving members | ||||||
| Diameter of flywheel $D$, mm, or drum, threshing cylinder | — | 790 | 880 | 900 | 810 | 1200 |
| Number of rev. per min. $n_x$ | 536 | 25.0 | 22.4 | 21.9 | 24.4 | 16.0 |
| Transmission ratio to the crankshaft $i'$ | $\dfrac{14}{14}\dfrac{19}{14}=1.36$ | $\dfrac{83}{12}\dfrac{46}{12}=26.6$ | $\dfrac{44}{10}\dfrac{49}{14}=12.3$ | $\dfrac{91}{14}\dfrac{45}{14}=18.5$ | $\dfrac{77}{14}\dfrac{37}{11}=18.5$ | $\dfrac{56}{10}\dfrac{34}{10}=19.05$ |
| Crank | ||||||
| Number of crank revolutions, $n$ | 727 | 665 | 262 | 455 | 452 | 305 |
| Angular velocity, $\omega$, sec.$^{-1}$ | 76.0 | 69.6 | 27.4 | 48.0 | 47.6 | 31.9 |
| Crank radius $r$, mm | 37.5 | 37.0 | 79.0 | 38.5 | 38.5 | 51.0 |
| Eccentricity $\varepsilon$, mm | −9.0 | −7.0 | +14.0 | −5.0 | −2.0 | +22.0 |
| Weight of crank $G_{kp}$, kg | 9.500 | 3.985 | 1.752 | 2.754 | 2.727 | 1.270 |
| Mass $m_{kp}$, $\dfrac{\mathrm{kg}\,\mathrm{sec}^{2}}{\mathrm{mm}}$ | 0.969 | 0.406 | 0.179 | 0.281 | 0.278 | 0.130 |
| Mass of crank reduced to the pin, $m_{kp}\dfrac{\varepsilon}{r}$ | −0.232 | −0.077 | +0.032 | −0.036 | −0.014 | +0.056 |
| Connecting rod | ||||||
| Length of connecting rod $l$, mm | 930 | 962 | 760 | 770 | 375 | 620 |
| Distance from pin to center of gravity $S$, mm | 420 | 432 | 382 | 276 | 123 | 310 |
| Weight of connecting rod $G_{sh}$, kg | 3.117 | 2.556 | 0.630 | 2.015 | 1.383 | 2.385 |
| Mass of connecting rod $m_{sh}$ | 0.318 | 0.261 | 0.064 | 0.205 | 0.141 | 0.243 |
| Part of the connecting-rod mass referred to the head, $m_{sh}\dfrac{s}{l}$ | 0.144 | 0.118 | 0.032 | 0.073 | 0.046 | 0.122 |
Explanations to Table No. 1
| Group | Quantity | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Connecting rod | Part of the mass of the connecting rod referred to the crank pin, \(m_{\mathrm{ш}}\dfrac{l-s}{l}\) | 0.174 | 0.143 | 0.032 | 0.073 | 0.046 | 0.122 |
| Connecting rod | Moment of inertia with respect to the center of gravity, \(\Theta_a\), kg·sec\(^2\)·m | 0.0450 | 0.0398 | 0.0050 | 0.0145 | 0.0025 | 0.0082 |
| Connecting rod | Radius of inertia with respect to the center of gravity, \(\rho_0\) | 376 | 390 | 279 | 266 | 134 | 184 |
| Connecting rod | Moment of inertia with respect to the crank pin, \(\Theta_a\) | 0.101 | 0.0886 | 0.0143 | 0.0301 | 0.0047 | 0.0315 |
| Connecting rod | Radius of inertia with respect to the crank pin, \(\rho_a\) | 563 | 582 | 473 | 383 | 182 | 360 |
| Knife | Weight of the knife, \(G_{\mathrm{н}}\) | 4.860 | 3.386 | 4.115 | 2.937 | 3.369 | 13.162 |
| Knife | Mass of the knife, \(m_{\mathrm{н}}\) | 0.494 | 0.345 | 0.419 | 0.299 | 0.343 | 1.342 |
| Whole mechanism | Rotating mass | \(-0.058\) | \(+0.066\) | \(+0.064\) | \(+0.096\) | \(+0.081\) | \(+0.177\) |
| Whole mechanism | Mass moving translationally | 0.638 | 0.463 | 0.451 | 0.372 | 0.389 | 1.464 |
| Whole mechanism | \(\lambda=\dfrac{r}{l}\) | 0.0403 | 0.0385 | 0.104 | 0.050 | 0.1026 | 0.0822 |
| Whole mechanism | \(k=\dfrac{a}{r}\) | 8.80 | 8.11 | 1.27 | 3.89 | 1.69 | 0.216 |
| Whole mechanism | \(\lambda_k=\dfrac{a}{l}\) | 0.357 | 0.312 | 0.132 | 0.195 | 0.174 | 0.0178 |
| Whole mechanism | \(a=rk\) | 0.330 | 0.300 | 0.100 | 0.150 | 0.065 | 0.011 |
| Whole mechanism | \(\dfrac{1}{\lambda}=\dfrac{l}{r}\) | 24.8 | 25.67 | 9.61 | 20.0 | 9.74 | 12.2 |
| Whole mechanism | \(\lambda_k^2=\dfrac{a^2}{rl}\) | 3.14 | 2.53 | 0.167 | 0.758 | 0.293 | 0.0038 |
| Whole mechanism | \(\omega^2 r\), m/sec\(^2\) | 216.6 | 180 | 59.2 | 88.7 | 87.3 | 51.9 |
- The number of revolutions of the running wheel per minute is determined by the formula
\[ n_x=\eta\,\frac{V\cdot 60}{\pi\cdot D}\ldots\ldots\ldots\ldots\ldots\ldots (30) \]
where \(\eta\) is the coefficient of wheel slip; according to test data it fluctuates within the limits \(0.95\text{--}0.84\),
on average \(\eta = 0.94\).
\(V\) is the translational speed of the machine, in meters per second.
(for a horse, \(V = 1.1\ \text{m/sec}\))
\(D\) is the diameter of the running wheel.
-
The position of the center of gravity of the connecting rod is determined by placing the connecting rod on a knife edge.
-
The position of the center of gravity of the crank is determined by suspension in several different positions.
-
The moment of inertia of the connecting rod is determined by suspending it on a prism by the bushing of the crank pin; if \(G_{\text{sh}}\) is the weight of the connecting rod in kg,
\(m_{\text{sh}}\) is its mass,
\(n\) is the number of complete oscillations per second,
\(h\) is the distance from the edge of the prism to the center of gravity of the connecting rod, then
the moment of inertia of the connecting rod relative to the edge is determined by the formula
\[ \Theta_{\text{edge}}=\frac{G_{\text{sh}}h}{4\pi^2 n^2}\ . . . . . . . . . . . . . . (31) \]
And the moment of inertia relative to the center of gravity is
\[ \Theta_o=\Theta_{\text{edge}}-m_{\text{sh}}h^2\ . . . . . . . . . . . . . . (32) \]
Then the moment of inertia relative to the axis of the crank pin is determined:
\[ \Theta_a=\Theta_o+m_{\text{sh}}S^2\ . . . . . . . . . . . . . . (33) \]
Proceeding directly to the determination of the inertia forces and their moments, for the purpose of the most visual comparison of their magnitudes in various types of harvesting machines, the calculation is made on the assumption that the mass of the crank is equal to zero. This assumption will introduce a small error, since the crank, designed with considerations of strength, will have insignificant dimensions and weight. In reality, however, the cranks of most machines are more or less balanced, and therefore, if their weight is taken into account, the true state of affairs will be expressed somewhat unclearly.
Considering Figs. 2--7, where the inertia forces and their moments for unbalanced crank mechanisms are represented graphically, we see that they reach the magnitudes shown in Table 2.
Table No. 2.
Maximum magnitude of inertia forces and their moments.
| Quantity | Mower | Mower | Mower | Mower | Mower | Mower |
|---|---|---|---|---|---|---|
| Quantity | Tractor mower | Horse mower | Holt combine | Reaper | Self-dump header | Sheaf binder |
| Horizontal inertia force \(I_x\), kg | 187.5 | 115.2 | 88.3 | 46.3 | 45.7 | 31.4 |
| Vertical inertia force \(I_z\), kg | 37.7 | 25.7 | 6.3 | 11.7 | 8.3 | 1.9 |
| Moment \(M_{\mathrm{in}}\), kg m | 14.1 | 8.6 | 1.3 | 1.8 | 0.5 | 0.4 |
And consequently, in sequence, from the point of view of the magnitude of the arising inertia forces, we must arrange the harvesting machines in the following series: tractor mower, horse mower, combine, reaper, self-dump header, sheaf binder.
The influence of the inertia of the masses of crank-and-connecting-rod mechanisms on the machine proceeds in the following directions.
- All elements of the connecting-rod-and-crank mechanism receive additional stresses both within themselves and in the joints.
For example, the inertia forces developing in the knife alternately compress it and stretch it. Let us try to calculate these stresses.
Obviously,
\[ k_z=\frac{P}{F}, \]
but the force \(P\) will be equal to the maximum value of the inertia force determined by formula (23), which approximately will be equal to
\[ P=m_n\omega^2 r. \]
If the dangerous section is taken to be the section passing through the last rivet fastening the back of the knife to the heel, then
\[ m_n=\frac{Fl\delta+nq}{g}, \]
where—\(F\)—section of the knife back in \(\mathrm{cm}^2\)
\(l\)—length of the back in cm, measured to the left from the dangerous section,
\(\delta\)—specific weight of the back material
\(n\)—number of segments located to the left of the dangerous section
\(q\)—weight of one segment in kg
\(g\)—acceleration of gravity.
In turn, the transverse section of the back subjected to the load is
\[ F=(a-d)\Delta,\ldots\ldots\ldots (34) \]
where—\(a\)—the width of the back in cm,
\(d\)—the diameter of the rivet in cm,
\(\Delta\)—the thickness of the back in cm.
The section \(F\) entering into the expression for the mass of the knife will be taken as
\[ F = a\Delta . \]
Substituting the expressions obtained into the stress formula, we obtain:
\[ K_z = \frac{\omega^2 r}{g}\cdot \frac{a\Delta l\delta + nq}{(a-d)\Delta} \ldots \ldots \ldots (35) \]
The tensile stresses calculated by this formula are shown in Table No. 3.
Table No. 3.
Tensile forces due to inertia forces in the back of the knife.
| Machine | \(\omega^2 r\) | \(g\) | \(a\) | \(\Delta\) | \(d\) | \(l\), total | \(l\), dangerous section | \(\delta\) | \(n\), total | \(n\), dangerous section | \(q\) | \(k_z\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Tract. mower | 216.6 | 9.81 | 2.0 | 0.6 | 0.5 | 225 | 191 | 0.0079 | 28 | 25 | 0.066 | 92 |
| Horse-drawn mower | 180 | — | 2.0 | 0.6 | — | 145 | 122 | — | 18 | 16 | 0.066 | 70 |
| Combine | 59.2 | — | 2.5 | 0.6 | — | 644 | 600 | — | 60 | 60 | 0.065 | 55 |
| Lobogreyka | 88.7 | — | 2.0 | 0.5 | — | 165 | 140 | — | 20 | 18 | 0.053 | 25 |
| Reaper | 87.3 | — | 2.2 | 0.5 | — | 167 | 142 | — | 20.5 | 18.50 | 0.053 | 23 |
| Sheaf-binder | 51.9 | — | 2.2 | 0.5 | — | 215 | 182 | — | 23 | 23 | 0.061 | 19 |
The stresses obtained, as we see, are not so great that one need be especially concerned about them.
However, from formula (34) it is evident that the tensile stresses in the knife increase directly in proportion to the square of the angular velocity of the crank, and therefore, if we were, for example, to increase the angular velocity of the combine crank fivefold, i.e., make \(n = 1515\), then the stresses would increase twenty-fivefold, i.e., up to 1375 kilograms per square centimeter.
-
Owing to the looseness present at the points where the elements of the mechanism are connected with one another, impacts occur, further enlarging the existing clearances and causing rapid wear of all the joints of the mechanism.
-
Since, in the motion of any system, the common center of gravity tends to remain in place, the displacement of the elements of the connecting-rod-and-crank mechanism in one direction will correspond to a displacement of the remaining parts of the machine in the opposite direction. And since the crank-and-connecting-rod mechanism performs a complex reciprocating motion combined with rotary motion, the entire machine likewise performs oscillatory motion, both linear and rotary.
Because every harvesting machine, in one way or another, moves over the ground, the oscillatory motion of the whole machine as a whole is transmitted through the wheel lugs into the soil and there, producing various deformations, both elastic and plastic, is damped.
The magnitude of the oscillations of the entire machine as a whole will obviously depend on the size of the machine itself or, more precisely, on the mass of the machine and on the moment of inertia of this mass with respect, for example, to the center of gravity. Therefore, from the magnitude of the inertial forces it is impossible to say how harmful they are in one machine or another: it is always necessary to take into account the weight of the machine, the form of the organs of adhesion to the soil, the distribution of weight with respect to the points of support, etc.
Up to now these questions have not yet been studied in such detail that the question of the character of the machine’s oscillation could be analyzed theoretically. In any case, for the present it may be noted that the oscillatory motion of the machine will be greater: 1) the greater the inertial forces of the connecting-rod crank mechanisms and their moments, 2) the smaller the total weight of the machine, 3) the closer together the points of support are located, 4) the fewer points of support the machine has.
All these factors are the reason why, in practice, the effects of mass inertia prove most harmful in mowers, then in reapers and headers.
In combines and sheaf binders they have very little effect on the operation of the whole machine.
IV. BALANCING.
The problem of balancing inertial forces and moments must obviously consist in eliminating the inertial forces and moments arising in the connecting-rod crank mechanism, or, in the worst case, reducing them to such an extent that in practice their action is not especially harmful. Let us see to what extent this is possible under the conditions of harvesting machines of existing types.
A. Complete balancing.
Complete balancing of all inertial forces and their moments would be possible only with the simultaneous presence of the following conditions, directly following from equations 27, 28, 29:
$$ m_{\sh}\frac{s}{l}+m_{\n}=0 \qquad \ldots\ldots\ldots\ldots (36) $$
$$ m_{kr}\frac{\varepsilon}{r}+\frac{l-s}{l}\,m_{\sh}=0 \qquad \ldots\ldots\ldots (37) $$
$$ ls-\rho_a^2=0 \qquad \ldots\ldots\ldots\ldots (38) $$
From (36) we have
$$ m_{\sh}=-\frac{l}{s}m_{\n} \qquad \ldots\ldots\ldots\ldots (39) $$
But the mass of the knife \(m_{\n}\) and the length of the connecting rod \(l\) are essentially positive, and \(m_{\sh}\) is also. Consequently, in order that the right-hand side be positive, it is necessary that \(s\) be negative; otherwise—
... speaking, the center of gravity of the connecting rod must be located not between points \(A\) and \(B\), but beyond point \(A\) (see Fig. No. 8).
In exactly the same way, from (37),
\[ m_{cr} = -\frac{r}{\varepsilon}\cdot\frac{l-s}{l}m_{sh}. \]
Substituting (39) here, we obtain:
\[ m_{cr}=\frac{r}{\varepsilon}\left(\frac{l}{s}-1\right)m_{n}. \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (40) \]
Here, in exactly the same way, \(\varepsilon\) must be negative, since \(s\) is negative, while the remaining quantities are essentially positive. Thus, the center of gravity of the crank must be, relative to the crank pin, on the side opposite the axis of the crankshaft (see Fig. No. 8).
If we were able practically to satisfy conditions (39) and (40), we would thereby completely eliminate the inertia forces; however, owing to the negative \(s\), (38) cannot in any way be satisfied, in other words, the moments of the inertia forces cannot disappear.
Conversely, with positive \(s\), by a rational design of the connecting rod it would be possible to satisfy condition (38) and eliminate the moments; but thereby we would make it impossible to eliminate the inertia forces.
Consequently, by means of counterweights alone it is impossible to eliminate simultaneously both the inertia forces and their moments.
Let us see what masses the connecting rod and the crank must have in order to completely balance the inertia forces.
As an example, let us take the McCormick-Deering tractor mower.
If we take
\[ s=-0.100\ \mathrm{m} \]
\[ \varepsilon=-r=-0.0375\ \mathrm{m}, \]
then
\[ m_{sh}=-\frac{0.980}{-0.100}\,0.494=\sim 4.594\ \frac{\mathrm{kg}\cdot \mathrm{sec}^{2}}{\mathrm{m}} \]
\[ \text{weight }G_{sh}=45.0\ \mathrm{kg}. \]
\[ m_{cr}=\frac{-0.0375}{0.0375}\left(\frac{0.930}{-0.100}-1\right)0.494 =\sim 5.088\ \frac{\mathrm{kg}\cdot \mathrm{sec}^{2}}{\mathrm{m}} \]
\[ \text{weight }G_{cr}=50.0\ \mathrm{kg}. \]
Thus, for complete balancing of the inertia forces in a tractor mower, the connecting rod and the crank turn out to be so cumbersome that there is no point even in speaking of their structural design.
Nevertheless, from the theoretical point of view we are interested in the question of what magnitude the moment of the inertia forces reaches under such balancing.
For this we need to know the radius of inertia of the mass of the connecting rod with respect to the crank pin. Since we shall not know it until we have designed such a connecting rod, in order to simplify the problem let us assume that the entire mass of the connecting rod is concentrated at its center of gravity; then, obviously,
\[ \rho_a = |s| = 0.100 \text{ m}. \]
Figure No. 9 shows graphically the variation of the moment of the inertia forces over one revolution of the crank.
It follows that, having balanced the inertia forces, we obtain a moment so large that it is difficult to say which is better: large inertia forces and a small moment, or, conversely, the absence of inertia forces and a very large moment.
Thus, complete balancing of the inertia forces by means of counterweights is impracticable both because of design difficulties and because it entails an increase of the moments to a magnitude that is almost equivalent to the eliminated forces.
B. Balancing of rotating masses.
In view of the fact that complete balancing of the crank-and-connecting-rod mechanism is impossible because of a large increase in the moments, one may set a much narrower task and balance only the inertia forces of the rotating masses. These, obviously, include the crank and the corresponding part of the connecting rod. Obviously, in this case it is necessary to satisfy the condition following from formula (28):
\[ m_{kr}\frac{e}{r} + \frac{l-s}{l}m_{sh} = 0 \ldots \ldots \ldots (41) \]
whence
\[ m_{kr}\frac{e}{r} = -\frac{l-s}{l}m_{sh} \ldots \ldots \ldots (42) \]
or
\[ m_{kr} = -\frac{r}{e}\cdot\frac{l-s}{l}\cdot m_{sh} \ldots \ldots \ldots (42a) \]
That is, the center of gravity of the crank must lie on the extension of the crank radius in the direction opposite to the pin.
In this case the following will remain unbalanced:
\[ I_x = -\omega^2 r \left[m_{sh}\frac{s}{l}+m_n\right] \left[\Cos\varphi + \lambda k \Sn\varphi - \lambda\Cos 2\varphi\right] \]
\[ M_{in}=m_{sh}\omega^2 r\,\frac{ls-\rho_a^2}{l} \left[ \lambda k\Cos 2\varphi - \left( \frac{1}{\lambda}+\frac{\lambda k^2}{2}-\lambda+\frac{3}{2}\lambda\Sn^2\varphi \right)\Sn\varphi \right]. \ (43). \]
For example, for the M.K.D. 7′ mower, if the same eccentricity is retained, the crank mass necessary for balancing the rotating masses is
\[ m_{kr}=-\frac{0.318}{-0.009}\cdot\frac{0.930-0.420}{0.930}\cdot 0.494 =0.72\ \frac{\mathrm{kg}\cdot \mathrm{sec}^{2}}{\mathrm{m}} \]
or
\[ G_{kr}=7.03\ \mathrm{kg}. \]
The inertia forces and moment that remain unbalanced in this case are shown in Fig. No. 10, from which it is evident that the vertical forces have disappeared, while the horizontal forces and the moment have decreased.
B. Balancing in existing machines.
In existing machines, the inertia forces of the connecting-rod–crank mechanism are usually balanced only by means of a single counterweight on the crank.
Therefore one can speak only of the extent to which the inertia forces of the rotating masses are balanced. The answer to this question is given by Table No. 4.
Table No. 4.
| Tractor mower M.K.M. | Horse mower G.L.Z. | Combine Holt 36 | Header, Krasny “Aksai” | Self-rake reaper “Kommunar” | Horse binder G.L.Z. | |
|---|---|---|---|---|---|---|
| \(m_M \dfrac{l-s}{l}\) | −0.174 | −0.143 | −0.122 | −0.073 | −0.046 | −0.032 |
| \(m_{kr}\dfrac{e}{r}\) | −0.232 | −0.077 | +0.056 | −0.036 | −0.014 | +0.032 |
| Is there balancing | Yes | Yes | No | Yes | Yes | No |
| By what % are the rotating masses balanced | 133% | 51% | — | 49% | 30% | — |
From this table we see that only one tractor mower, the MKD 7′, has a counterweight greater than is needed for balancing the rotating masses alone by approximately 1/3. Because of this, as Fig. No. 11 shows, vertical inertia forces reappear, but with opposite signs.
The horizontal inertia forces are reduced by the corresponding amount.
Thus it must be acknowledged that, if we generally wish to reduce the inertia forces by means of a counterweight on the crank, it must be designed so that equation (42) is satisfied. From this point of view, the counterweights of the horse mower, the header, and the reaper are insufficient.
A further increase of the counterweight cannot eliminate the forces of inertia, but only transfers the horizontal forces into vertical ones and increases the moment; therefore it can be considered useful only when, taking into account the design of the given machine, one may be confident that the vertical forces are less dangerous for it than the horizontal ones, or at least equivalent to them.
References
- V. P. Goryachkin. Forces of inertia and their balancing.
- B. A. Kril. Some data from tests of mowers at M.I.S.T.S.Kh.A. in 1926–1927, Bulletin of the Metal-Processing Industry for 1929.
- E. D. Lvov. Tractors, their design and calculation.
- G. Lorenz. Technical mechanics of an invariable system.
- M. N. Letoshnev. The cutting apparatus of harvesting machines. Lecture notes.
- Yu. A. Weiss. Course in agricultural machinery science.
SUMMARY.
To the Question of Counterbalancing the Connecting Rod-Crank Mechanism of Harvesting Machines.
By Eng. G. D. Terskov.
- Denoting by:
$m_{kp}$ — the mass of the crank,
$m_{\mathrm{sh}}$ — the mass of the connecting rod,
$m_k$ — the mass of the sickle knife,
$\omega$ — the angular speed of the crankshaft,
$r$ — the radius of the crank,
$\varepsilon$ — the distance from the centre of gravity of the crank to the rotation axis,
$l$ — the length of the connecting rod,
$s$ — the distance from the center of gravity of the connecting rod to the crank pin,
$a$ — the displacement of the crankshaft axle from the line of movement of the sickle knife,
$\rho_a$ — the radius of inertia of the connecting rod with regard to the crank pin,
\[ \lambda=\frac{r}{l}, \qquad k=\frac{a}{r}, \]
$\varphi$ — the angle of the turn of the crank measured from the right horizontal position (the machine seen in front),
we find that the value of the forces of inertia of the whole connecting rod-crank mechanism are determined by the expressions:
\[ J_x=\omega^2 r\left[\left(m_{kp}\frac{\varepsilon}{r}+m_{\mathrm{sh}}\frac{l-s}{l}\right)\Cos\varphi+ \left(m_k+m_{\mathrm{sh}}\frac{s}{l}\right) \left(\Cos\varphi+\lambda k\,\Sn\varphi-\lambda\Cos 2\varphi\right)\right] \tag{27} \]
\[ J_z=\omega^2 r\left(m_{kp}\frac{\varepsilon}{r}+m_{\mathrm{sh}}\frac{l-s}{l}\right)\Sn\varphi \ . . . . . . . \tag{28} \]
Besides this, owing to the inequality of the rotatory movement of the connecting rod, the inertia couple will take place:
\[ M_{\mathrm{in}}=-\omega^2 r\left\{\left(m_{kp}\frac{\varepsilon}{r}+m_{\mathrm{sh}}\frac{l-s}{l}\right)a\Cos\varphi -m_{\mathrm{sh}}\frac{ls-\rho_a^2}{l} \left[\lambda k^2\Cos 2\varphi- \left(1+\frac{\lambda^2 k^2}{2}-\lambda^2+\frac{3}{2}\lambda^2\Sn^2\varphi\right)\Sn\varphi\right]\right\} \ . . . \tag{29} \]
But, \(\omega^{2} r \operatorname{Cos}\varphi\) and \(-\omega^{2} r \operatorname{Sn}\varphi\) being terms which determine the acceleration of the movement of the crank pin, and
\[ \omega^{2} r\left(\operatorname{Cos}\varphi+\lambda_k \operatorname{Sn}\varphi-\lambda \operatorname{Cos}2\varphi\right) \]
those determining the acceleration of the movement of the sickle knife, it appears from formulae (27) and (28) that:
a) when determining the forces of inertia, the whole connecting rod–crank mechanism may be substituted by two masses;
b) the first mass is concentrated on the crank pin and rotates equably about the axle of the crankshaft and is determined by the expression
\[ \left[m_{kp}\frac{\varepsilon}{l}+m_{\sh}\frac{l-s}{l}\right]; \]
c) the second mass is concentrated on the head of the knife, has a reciprocating movement and is determined by the expression:
\[ \left[m_{\sh}\frac{s}{l}+m_k\right]. \]
In other words, when determining the forces of inertia of the connecting rod–crank mechanism, the mass of the crank may be considered as distributed on the crank pin and on the head of the knife in inverse proportion to the distances of the center of gravity of the connecting rod from these points.
- The influence of the forces of inertia of the connecting rod-crank mechanism upon the machine is expressed as follows:
a) All the parts of the connecting rod-crank mechanism get additional strains the value of which increases in direct proportion to the square of the angular speed of the crank.
b) There occur impacts in the joints which increase their wear. The whole machine performs an oscillating movement partly transmitted into earth through the spurs of wheels and damped there. The oscillating movement of the machine will increase the greater are the forces of inertia of the connecting rod-crank mechanism and the less is the total weight of the machine and the number of points of application.
- A complete counterbalancement of all forces of inertia and of their moments would be possible only by a simultaneous accomplishment of conditions issuing immediately from (27), (28) and (29):
\[ m_{\sh}\frac{s}{l}+m_k=O \ldots\ldots\ldots\ldots (36) \]
\[ m_{kp}\frac{\varepsilon}{r}+\frac{l-s}{l}m_{\sh}=O \ldots\ldots\ldots (37) \]
\[ l_s-\rho_a^2=O \ldots\ldots\ldots\ldots (38) \]
But since (36) can be fulfilled only when \(s\) is a negative value, and (38), on the contrary, only when \(s\) is a positive value, it is clear that it is not possible to counterbalance the forces of inertia as well as those of their moments simultaneously by means of counterbalances.
The counterbalancing only of the forces of inertia by means of counterweights is quite possible. It is only necessary to have on the crank a counterweight with a mass:
\[ m_{kp}=-\frac{r}{\varepsilon}\left(\frac{l}{\delta}-1\right)m_{h} \tag{40} \]
and on the connecting rod a counterweight with a mass:
\[ m_{ш}=-\frac{l}{\delta}m_{h}. \tag{39} \]
In practice, however, these counterweights prove to be very cumbersome (being 50 kg each for the tractor mowing machine), and besides this, there occurs a considerable increase of moments. It proves, thus, that the counterbalancing of all forces of inertia by means of counterweights cannot be put into practice owing to constructional inconveniences and is not effective enough owing to the increase of moments resulting from the forces of inertia.
The counterbalancing of the rotating masses only [in other words, of vertical forces of inertia] is quite practicable; in order to attain this it is necessary to set on the crank a counterweight having a mass:
\[ m_{kp}=-\frac{r}{\varepsilon}\frac{l-s}{l}m_{ш} \tag{42a} \]
If a counterweight with a mass greater than that given in formula (42a) is taken, it will cause a decrease of horizontal forces of inertia and a corresponding increase of vertical forces of inertia; therefore, in the construction of harvesting machines it is necessary to endeavor to fulfill condition [42a] and to admit the further increase of the counterweight only in those cases when it is possible to be quite certain that vertical forces of inertia are less ruinous for a given machine than the horizontal ones.
- Of the existing harvesting machines, the tractor mowing machine is the one where the forces of inertia reach their greatest significance, and the horse-binder is the machine where they are of least importance (this being obvious from table No. 2). Usually, counterweights having a mass less than that required for counterbalancing vertical forces of inertia are installed, and only in the Mc. Cormick Deering tractor mowing machine (the driving one) is a counterweight 33% greater than that required by formula (42) installed.
Fig. 1.
TRACTOR MOWER MKD-7.
Inertia of the masses of the connecting-rod–crank mechanism.
Fig. 2.
Horse-drawn mower, GLZ. 4½′
Inertia of the masses of the connecting-rod–crank mechanism.
Fig. 3.
Holt 36 combine.
Inertia of the masses of the connecting-rod–crank mechanism.
Fig. 4.
“KR. AKSAI” HEADER
Inertia of the masses of the connecting-rod–crank mechanism.
Fig. 5.
“KOMMUNAR” HEADER.
Inertia of the masses of the connecting-rod–crank mechanism.
Fig. 6.
HORSE-DRAWN SHEAF BINDER GIZ.
Inertia of the masses of the connecting-rod–crank mechanism.
Fig. 7.
Fig. 8.
TRACTOR MOWER MKD-7.
“Complete” balancing.
Fig. 9.
TRAJECTORY OF POINT 7.
Balancing mass.
Fig. 10.
TRAJECTORY OF POINT 7.
Inertia of the mass in reality.
Fig. 11.