V. G. Karpenko.

Balancing of Masses in Internal-Combustion Engines

The widespread use of high-speed internal-combustion engines in various branches of technology—in motor-ship construction, in the automobile field, in aviation, and so forth—has placed the question of balancing the masses in the first rank in the development of the designs of these machines. Meanwhile, in Russian technical literature there is as yet no work that treats this question with sufficient completeness, which is keenly felt by all those who have to deal with the design of internal-combustion engines. All existing courses on internal-combustion engines illuminate this question very inadequately, touching upon it only in passing and, moreover, sometimes presenting it incorrectly (H. Güldner*). In his work under the above title the author fills the existing gap in Russian technical literature, confining himself within the limits of the study of existing machines.

The question of the influence of moving masses on the running of piston machines first arose in relation to steamship engines, where it is as acute as it is in high-speed internal-combustion engines. There it was first raised as early as 1849 by Lechatelier, although the complete theory of the reciprocating motion of a steamship, in corrected form, was given by him in 1851. In 1863 the German engineer Schlick succeeded, after a series of experiments, in obtaining balanced steamship engines without resorting to the installation of counterweights, but by assigning only definite values to the angles between the cranks and to the distances between the mechanisms and the masses of the reciprocating parts. The disputes that arose over priority, which Taylor and Yarrow contested with Schlick, were the cause of further investigations of the question by Schubert in his Theorie des Schlikschen Massenausgleiches and by Lorenz in the works Dynamik der Kurbelgetriebe and Technische Mechanik starrer Systeme, which helped to clarify this question. Later we see the thorough works of Sharp, Balancing of engines, steam, gas, and petrol, and of Kölsch, Gleichgang und Massenkräfte bei Fahr— und Flugzeugmaschinen. The latter work, as its very title shows, relates exclusively to internal-combustion engines, and moreover to engines of a narrowly specialized purpose.

In March 1912 V. P. Arshaulov read a report in the Society of Technologists on the motor ship Borodino; a development of this report was an article by the same author, published in 1913 in the Bulletin of the Society of Technologists under the title: “On the Balancing of Inertia Forces in the Diesel Engine.” In this article V. P. Arshaulov gives a solution to the question of balancing marine internal-combustion engines, carried out in several machines built for motor ships by the Kolomna Works. This solution, as may be seen, coincides in principle with the solution of the question of balancing marine steam engines given in its time by Schlick, but is distinguished by extraordinary elegance and simplicity. A year later, in the same Society of Technologists and on the same subject, his

*) See the Russian translation of H. Güldner’s book, p. 230—1916.

Prof. A. A. Radtsig’s report, distinguished by simplicity and clarity of exposition. Prof. Radtsig presented the question approximately in the form in which it is set forth by Prof. Lorenz, and drew the meeting’s attention to the importance of the transverse moment caused by angular acceleration and undoubtedly having a substantial influence on the smooth running of machines.

The article by V. P. Arshaulov and the report by Prof. A. A. Radtsig, printed in the Bulletin of the Society of Technologists in 1913, made a deep impression on broad circles of engineers.

At the end of 1918, by resolution of the Mechanical Faculty, I was offered, for the diploma of engineer of the first degree, the topic: “Balancing of masses in internal-combustion engines according to V. Arshaulov and O. Kölsch.” The work submitted by Eng. Kvasnikov proved interesting not only in the sense of answering the question posed, but also from the standpoint of an attempt to give an initial formula for finding the free forces and moments. Its chief merit is undoubtedly the introduction of the angle of arrangement of the cylinders into the equations of the free forces and moments given by Lorenz; and although the general form of the equations is thereby complicated, the solution of special cases is simplified.

In carrying out an investigation of the question of balancing the masses of internal-combustion engines, I made use of the investigations and data of Lorenz, Sharp, Kölsch, Arshaulov, Radtsig, and the formula indicated by Eng. Kvasnikov. The investigation consisted chiefly in studying individual cases of balancing machines according to the principle of Pfützner–Arshaulov and by means of counterweights. The general formulas of the free forces and moments have too complicated a form for anyone wishing to acquaint himself with the properties of a machine to be able to orient himself quickly and easily by them. To solve the question in a particular case one often has to resort to lengthy calculations, which under practical conditions is inconvenient; therefore I give a summary of the results of the work, in the form of a separate table of solutions of particular cases.

General Form of the Equilibrium Conditions.

In all piston machines there act:

1) the inertia forces of the reciprocating parts and
2) moments connected by the general equation

\[ \sum m_p + \sum m_j + \sum m_e = 0, \]

where \(\sum m_p\) is the sum of the moments of the acting forces and of the resistance forces; \(\sum m_j\) is the sum of the moments of all inertia forces, both reciprocating and rotating at constant angular velocity; and \(\sum m_e\) is the sum of the moments of the inertia forces caused by the additional angular acceleration \(\frac{d\omega}{dt}\). The sums of moments \(\sum m_p\) and \(\sum m_e\) overturn the machine about the shaft axis; the sum of moments \(\sum m_j\) overturns the machine about an axis perpendicular to the plane passing through the shaft and the cylinder axes. In addition, there is also a transverse moment with respect to an axis parallel to the shaft axis, caused by the oscillating masses of the connecting rod.

Proceeding from the motion of the crank mechanism, one may derive, for the inertia forces of a multi-cylinder machine with the arrangement of cylinders according to Fig. 5, the following equations:

\[ X_0 = -mr\omega^2(\cos\psi_0 \sum \cos\varphi \cos\gamma - \sin\psi_0 \sum \sin\varphi \cos\gamma + \]

\[ + A_2 \cos 2\psi_0 \sum \cos 2\varphi \cos\gamma - A_2 \sin 2\psi_0 \sum \sin 2\varphi \cos\gamma \ldots) - \]

\[ - mr\dot{\omega}(\sin\psi_0 \sum \cos\varphi \cos\gamma + \cos\psi_0 \sum \sin\varphi \cos\gamma + \]

\[ + \tfrac{1}{2} A_2 \sin 2\psi_0 \sum \cos 2\varphi \cos\gamma - \tfrac{1}{2} A_2 \cos 2\psi_0 \sum \sin 2\varphi \cos\gamma \ldots). \]

\[ \begin{aligned} Y_0={}&-mr\omega^2\left(\cos\psi_0\sum \cos\varphi \sin\gamma-\sin\psi_0\sum \sin\varphi \sin\gamma+\right.\\ &\left.+A_2\cos 2\psi_0\sum \cos 2\varphi \sin\gamma-A_2\sin 2\psi_0\sum \sin 2\varphi \sin\gamma+\ldots\right)-\\ &-mrE\left(\sin\psi_0\sum \cos\varphi \sin\gamma+\cos\psi_0\sum \sin\varphi \sin\gamma+\right.\\ &\left.+\frac{1}{2}A_2\sin 2\psi_0\sum \cos 2\varphi \sin\gamma+\frac{1}{2}A_2\cos 2\psi_0\sum \sin 2\varphi \sin\gamma+\ldots\right). \end{aligned} \]

For vertical machines,

\[ \gamma=0,\quad \varphi=\alpha \quad \text{and} \quad Y_0=\sum m\frac{d^2Y}{dt^2}=0, \]

Consequently, in order to balance them it is necessary to satisfy the condition

\[ \sum \cos\alpha=0,\quad \sum \sin\alpha=0,\qquad \sum \cos 2\alpha=0,\quad \sum \sin 2\alpha=0. \]

Balancing of machines with respect to longitudinal moments.

If the distances between the axes of the cylinders are taken equal to \(z\), then, with respect to the inertia forces of the reciprocating masses, the moments may be composed:

\[ \begin{aligned} M_x={}&\sum mz\frac{d^2y}{dt^2} =-mr\omega^2\left(\sum z\cos\psi \sin\gamma +A_2\sum z\cos 2\psi \sin\gamma+\ldots+\right.\\ &\left.+A_n\sum z\cos n\psi \sin\gamma+\ldots\right) -mrE\left(\sum z\sin\psi \sin\gamma+\frac{1}{2}A_2\sum z\sin 2\psi \sin\gamma+\ldots\right.\\ &\left.+\frac{1}{n}A_n\sum z\sin n\psi \sin\gamma+\ldots\right). \end{aligned} \]

\[ \begin{aligned} M_y={}&\sum mz\frac{d^2x}{dt^2} =-mr\omega^2\left(\sum z\cos\psi \cos\gamma +A_2\sum z\cos 2\psi \cos\gamma+\ldots+\right.\\ &\left.+A_n\sum z\cos n\psi \cos\gamma+\ldots\right) -mrE\left(\sum z\sin\psi \cos\gamma+\frac{1}{2}A_2\sum z\sin 2\psi \cos\gamma+\ldots\right.\\ &\left.+\frac{1}{n}A_n\sum z\sin n\psi \cos\gamma+\ldots\right). \end{aligned} \]

The conditions of balancing will be

\[ \sum z\cos i\psi \cos\gamma=0,\quad \sum z\cos i\psi \sin\gamma=0, \]

\[ \sum z\sin i\psi \cos\gamma=0,\quad \sum z\sin i\psi \sin\gamma=0. \]

Without dwelling on the derivation of the equations of transverse moments and on the results of these derivations, I shall only point out that one of them, caused by the rocking masses of the connecting rod, gives equations generally similar to those indicated above and may be balanced by the method of Schlick–Arshaulov; the second moment, caused by angular acceleration, depends chiefly on the degree of non-uniformity of the machine.

When balancing forces and moments by means of counterweights, the balancing is carried out not to the full amount, but to one half of the greatest value of these forces and moments, since only under this condition is the best balance obtained. Proceeding from this proposition, I have calculated the masses of the counterweights in column IV of the table of research results; only in those cases where the magnitude and direction of the inertia force remain constant have I calculated the counterweights for complete balancing.

The table of successive results was obtained for various types of machines after substituting the values of the constant angles \(\phi\), \(\sigma\), and \(\gamma\) into the formulas given above. The following notation is adopted in the table: \(m\)—mass, \(m_k\)—mass of the reciprocating parts; \(m_b\), mass of the rotating parts; \(m_j\)—mass of the counterweights; \(r\)—crank radius; \(r_1\)—radius of the center of gravity of the counterweights; \(l\)—length of the connecting rod; \(a\)—distance to the shaft axis between the points of application of the inertia forces; \(a_k\)—distance between the inertia forces of the rocking masses; \(a_b\)—distance between ...

by the centrifugal forces of the rotating masses, \(a\) is the distance between the centrifugal forces of the counterweight, \(J_c\) is the moment of inertia of the connecting rod, reduced to the center of oscillation. The numerical coefficients of the expressions for the moments have been calculated on the assumption \(r:l = 1:5\); as for the coefficients \(A_2, A_4, \ldots\) and \(C_1, C_2, C_3, \ldots\), their values may be taken from the following table:

By substituting particular values of the quantities into the equations of the table of results, one can obtain the magnitudes of the free forces and moments, and also the magnitudes of the counterweights for balancing the machines.

The magnitude of the free forces and moments must, above all, be judged by their order; but one must not compare free forces of different orders, which differ not only in magnitude but also in frequency of action. Moreover, the absence of a free force is not yet an indication that the machine is balanced, since considerable moments are possible in this case, as we see, for example, in two-cylinder machines with cylinders and crankshafts arranged diametrically, or in paired star-shaped machines. In comparing machines by the magnitude of the order of the forces, one must not forget the existence of free moments, which play a more essential role, and also the possibility of their complete or incomplete balancing by counterweights. Thus, for example, star-shaped machines have a considerable free force of the first order, but one that is quite completely balanced by counterweights, which makes these machines suitable even for aircraft. The same may be said of fan-shaped machines, in which almost complete balancing is achieved by counterweights. As for vertical machines, four-cylinder machines with a mirror arrangement of the crankshafts, as well as six- and eight-cylinder machines, are well balanced. None of them requires counterweights, except for two-stroke eight-cylinder machines. Judging by the magnitude of the order of the free force, the most balanced machines will be six-cylinder ones. This explains their ever wider distribution in motor-boat building, aviation, and even in automobile engineering, where the four-cylinder machine, possessing sufficient balance for an automobile and being cheaper than a six-cylinder machine, has won such a firm position. In general, vertical multi-cylinder machines possess better balance than all other types of machines. In most cases they have not only no longitudinal moment, but also no transverse moment caused by the oscillating masses of the connecting rods; free forces occur only in higher orders. The reason for this is the symmetrical arrangement of the moving parts with respect to their common center of gravity, which cannot be attained to the same extent in other machine designs.

Engineer A. G. Kartenko.

Calculation of the distributing organs of two-stroke and four-stroke machines

Drawing 1.

Drawing 2.

Balancing of masses in internal-combustion engines

Drawing 3.

Drawing 1.

TABLE

of test results.

In the work of V. P. Karyachenko.