Assoc. Prof. V. K. NECHAEV

THEORETICAL TORSIOGRAMS

FOR THE SHAFT OF AN INTERNAL-COMBUSTION ENGINE

(On the problem of torsional vibrations of shafts)

United Scientific and Technical Publishing House
Sverdlovsk—Moscow
1937

I. INTRODUCTION

One of the most important dynamic characteristics of any power installation with a piston engine is the degree of nonuniformity of rotation of the main shaft, determined by the well-known formula:

\[ \delta=\frac{\eta_{\max}-\eta_{\min}}{\eta}, \tag{1} \]

where \(\eta_{\max}\) and \(\eta_{\min}\) are, respectively, the maximum and minimum angular velocities of rotation of the shaft (or, more precisely, of the given transverse section of it) during the cycle of the working process in the engine cylinders;

\[ \eta=\frac{\pi n}{30} \]

is the mean angular velocity of rotation of the shaft, corresponding to \(n\), the number of revolutions of the shaft per minute.

In a number of practical cases the specified value \(\delta\), required for production and operational reasons, determines the choice of engine type for the installation being designed.

Strict requirements with respect to the value of \(\delta\) are imposed, for example, on diesel-dynamo units for direct or alternating current operating on a lighting network. In such installations sufficient uniformity of rotation of the engine crankshaft must ensure the absence of flickering of the light.

But a small value of \(\delta\) cannot always serve as a guarantee of reliable and satisfactory operation of the installation from the dynamic point of view.

The practice and theory of the parallel operation of alternating-current diesel-generators show that what is essential here is not so much the magnitude of the degree of nonuniformity of rotation of the generator rotor as the angles of deviation of the rotor from the position which it would occupy at each instant of time under absolutely uniform rotation. If the angles of lag or lead of the rotor (relative to uniform rotation) are excessively large, very large equalizing currents arise in the generator-

... machines, there may be a falling-out of synchronism of the generators and, consequently, a complete disruption of the normal operation of the entire installation.

Therefore, for example, the rules and standards for alternating-current diesel generators operating in parallel impose very stringent requirements with respect to the magnitude of these angles and, to a considerably lesser extent, insist on a small value of the degree of nonuniformity of shaft rotation \((\delta)^2\).

Sufficient uniformity of running of a piston engine of a given type is usually achieved by an appropriate choice of the magnitude of the flywheel masses on the main shaft (above all, of the flywheel).

The well-known classical methods of calculating the flywheel (by the diagram of tangential forces or by Wittenbauer’s method), which are intended to solve the problem of the dynamic investigation of power installations with piston engines, give values of \(\delta\) that often diverge very greatly from reality (usually underestimated as compared with the actual values by \(1 \frac{1}{2}\)—2 times and more). Moreover, these methods of analyzing the nonuniformity of rotation do not make it possible to judge the magnitude of the lag or lead angles of the flywheel or the rotor of the dynamo with respect to its uniform rotation and therefore, at least in their usual form, are unsuitable, for example, for calculating the flywheels of alternating-current machines operating in parallel.^3

The cause of the above-mentioned considerable error in the calculation of \(\delta\) by the diagram of tangential forces or by Wittenbauer’s method should be considered, first of all, to be the assumption, concealed in both these methods, of the absolute torsional rigidity of the shaft of the installation.^4 This assumption underlies the supposition of the mutual equality of the angular velocities of rotation of all the masses located on the shaft at every instant of time.

In fact, however, owing to the compliance and elasticity of the shaft under the action of variable torques upon it and the consequent inevitability of its torsional vibrations, each

^1 H. Görges, Elektrotechnische Zeitschrift, 1900, p. 186; 1902, p. 1053 and following.
E. Rosenberg, E. T. Z., 1902, pp. 425, 450, 468.
    »             »     1904, pp. 793, 856.
F. Klönne           »     1902, p. 287 and following.
G. Lyust, Applicability of internal-combustion engines at electric stations, Proceedings of the Petrograd Polytechnic Institute, 1916, vol. XXV, pp. 1—145.
V. Ivanov, Synchronous Machines, 1934.

^2 V. K. Zhitomirskii, Parallel operation of diesel generators, Diesel Engineering, 1935, No. 1.
N. G. Kalish, The relation between the true degree of nonuniformity and the permissible deviation of the rotor in electrical degrees, Diesel Engineering, 1936, No. 2.

^3 F. Sass, Gleichgang von Kolbenkraftmaschinen, Archiv f. Wärmewirtschaft u. Dampfkesselwesen, 1928, p. 100, as well as the works cited in footnote 1.

^4 In this respect Wittenbauer’s method has no advantage over the calculation by the diagram of tangential forces.

of the masses rigidly connected with the shaft has its own angular velocity of rotation, varying with time. This leads to the fact that each mass and each section of the shaft has its own degree of nonuniformity of rotation, depending on the magnitude and arrangement of the masses of the elastic system, the location of the given mass (or the given section) on the shaft, the number of revolutions of the engine, the mode of its operation, etc.

The degree of nonuniformity \(\delta\) of the flywheel or dynamo rotor of a given installation is different at different rotational speeds of the engine shaft. It becomes especially large when the engine operates at critical (or close to critical) numbers of revolutions.

The dependence of \(\delta\) on \(n\) in reality proves to be quite different and considerably more complex than is established by the usual formula:

\[ \delta=\frac{3600A_s}{n^2GD^2}, \tag{2} \]

when calculated from the tangential-force diagram, or similarly, when calculated by Wittenbauer.

In formula (2):

\(A^s\) — the excess work absorbed by the flywheel.
\(GD^2\) — the flywheel moment of the flywheel.

Torsional vibrations of the shaft occur (to a greater or lesser extent) at any speeds of its rotation and under any operating modes of the engine; therefore the degree of nonuniformity of rotation \(\delta\), calculated by the classical methods of flywheel calculation, in essence never corresponds to reality.^5

But with accuracy sufficient for practice, this correspondence may exist only when there are very weak torsional vibrations of the shaft. Only in these cases (a low number of revolutions of the engine, a very short and comparatively thick shaft, a small magnitude of flywheel masses) do the usual methods of flywheel calculation give sufficiently reliable results. However, even under such conditions these methods of calculation are unsuitable for analyzing the uniformity of rotation of alternating-current diesel generators operating on a common network, owing to the appearance here of additional oscillations, the so-called “swinging” of the rotors.^6

The investigation of oscillations in similar installations, taking account of synchronizing moments, etc., lies beyond the scope of the present work. In all that follows we shall assume—

^5 K. Rerikh, Theory of Regulation of Machines, Part 1, p. 175, 1916.
V. Zhitomirskii, “The True Degree of Nonuniformity,” Dizeleostroenie, 1934, No. 1.
N. Pines, “Flickering of Light in Electrical Networks Due to Torsional Vibrations of the Rotors of Electrical Machines,” Elektrichestvo, 1931, pp. 193–201.
N. Makarevich, Installations of Internal-Combustion Engines, 1935.
F. Sass, “Arch. f. Wärmewirtschaft u. Dampfkesselwesen,” 1928, p. 100.

^6 See, for example, the works indicated in footnote 1, and also V. Mazuga, “Natural Frequency of Oscillations of a Diesel Generator during Parallel Operation with a Powerful Network,” Elektrichestvo, 1934, No. 6.

to regard the single operation of a diesel-generator set or the operation, for example, of a ship installation with a propeller.

The mechanical application of flywheel calculations by the diagram of tangential forces, or according to Wittenbauer, to all cases of practice may sometimes lead not only to inaccurate but also to entirely erroneous conclusions. The selection of a very large flywheel with a large \(GD^2\), ensuring, according to formula (2), a small value of \(\delta\), by no means always guarantees reliable operation of the power installation from the standpoint of sufficient uniformity of shaft rotation. In practice, cases have been observed in which the uniformity of rotation worsened when one flywheel was replaced by another with a larger \(GD^2\).¹ Sometimes a reduction of \(GD^2\), or even the complete removal of the flywheel, is the simplest means of improving the uniformity of shaft rotation at a given number of its revolutions.

Facts of this kind, quite paradoxical from the standpoint of the classical theory of the flywheel, receive an extremely simple and definite explanation in the theory of torsional vibrations of shafts. An analysis of these facts on the basis of the theory of vibrations leads to the concept of harmful flywheel masses and of the critical value of the flywheel \(GD^2\).²

The actual laws of rotation of individual sections of the main shaft of a power installation are represented graphically by torsiograms, which can be taken from a running engine with a Geiger, DVL, and other torsiographs. Each such torsiogram makes it possible to determine the actual degree of nonuniformity of rotation of the corresponding shaft section.

Similar torsiograms can be constructed with sufficient reliability by theoretical means, already during the design of the power installation.

At present, in both Russian and foreign literature, there is no sufficiently developed method for constructing theoretical torsiograms. Individual works on torsional vibrations of shafts, devoted in part or in full to this question,³ set forth only simplified methods for constructing torsio—

…torsiograms, based on a whole series of assumptions, for example: the entire installation is reduced to a two- or three-mass system, only the principal orders of the harmonics of the torque on the engine shaft are decisive, the engine operates outside the critical speeds, etc.

The task of the present work is to set forth a more detailed method for constructing theoretical torsiograms, based on the smallest possible number of assumptions, as well as methods for determining the most advantageous value of the flywheel moment of the flywheel.

The motion of each mass located on the shaft of a power installation is composed of the uniform rotation of the entire shaft as one rigid body with constant speed (equal to the mean angular speed of rotation of the shaft over a cycle in the cylinder) and of forced oscillatory motions superposed on this rotation, produced by the harmonic components of the torque on the engine shaft.

Each of these forced oscillatory motions has its own phase, amplitude, and frequency, depending on the law of the torque diagram, the characteristics of the reduced elastic system, the magnitude of the damping forces, the speed of rotation of the shaft, the order of alternation of flashes in the engine cylinders, etc.

The computational work for constructing a theoretical torsiogram for a given section of the shaft or for a given mass of the installation is divided into the following basic stages:

1) harmonic analysis of the diagram of torques on the engine shaft produced by one cylinder;

2) determination of the amplitudes of the forced oscillations of the given shaft section caused by the individual component harmonics of the torque;

3) summation of all forced oscillations of the given shaft section, i.e. construction for it of the theoretical torsiogram.

Before carrying out these items, it is necessary to perform the usual calculation of the installation for free torsional vibrations, determine the frequencies of the natural vibrations, and compute the critical numbers of revolutions of the engine shaft. In the event that any (especially principal) critical numbers of revolutions coincide with the operating speeds of the engine, measures must be taken to eliminate the dangers of the resonance arising in this case⁴. A final judgment on the possibility or inadmissibility of engine operation at certain critical speeds (usually secondary, of high orders) can be established only after calculating the additional stresses in the shaft.

The torque on the engine shaft is created by the pressure of gases in the cylinders, by the inertia forces of reciprocating masses, by the moments of the inertia forces of the connecting rods, and by the weight of the parts of the crank-and-connecting-rod mechanisms.^11

Harmonic analysis of the diagram of tangential forces, constructed from the indicator diagram without taking into account inertia forces, etc., makes it possible to represent the torque created on the engine shaft by the gas pressure in one cylinder in the form of a Fourier series:

\[ M' = M_0 + \sum M'_h \cdot \sin (h\eta t + \beta'_h), \tag{3} \]

where: \(M_0\)—the mean torque on the shaft over one operating cycle of the cylinder;

\(\eta\)—the angular velocity of rotation of the shaft;

\(h\)—the order of the harmonic of the torque with respect to one revolution of the engine shaft;

\(\beta'_h\)—the phase angle of the given harmonic of order \(h\);

\(M'_h\)—the amplitude of the harmonic.

In four-stroke engines, the order \(h\) of the harmonic components of the torque successively takes the values:

\[ h = \frac{1}{2};\quad 1;\quad 1\frac{1}{2};\quad 2 \ldots ; \]

in two-stroke engines:

\[ h = 1;\quad 2;\quad 3 \ldots \]

Each harmonic moment \(M'_h \cdot \sin (h\eta t + \beta'_h)\) may be represented as the projection of a rotating vector of length \(M'_h\) on the vertical (Fig. 1). The angular velocity of rotation of this vector is equal to \(h\eta\). The phase angle \(\beta'_h\) is the angle made by the vector \(M'_h\) with the axis of abscissae at the initial instant of time, at \(t = 0\). In all subsequent exposition we assume that the vectors of the moments rotate counterclockwise and that time is counted from the moment when the crank passes through the upper dead point.

^11 The torque produced on the shaft by the inertial forces of the connecting rod is only approximately taken into account by the usual decomposition of the connecting rod into rotating and reciprocating masses.

For an exact accounting of the influence of the connecting rod, see I. Neiman, Dynamics and Strength Calculation of Aircraft Engines, Part 1, 1933.

L. Martens, Dynamics of Piston Engines, 1932.

A. Gorfinkel, “Vibration de torsion dans les moteurs polycylindriques,” Génie Civil, 1928, p. 513.

F. Porter, “The range and severity of torsional vibration in Diesel engines,” Trans. A. S. M. E., Vol. 50, 1927—1928.

The influence of the weight of the parts of the crank-and-connecting-rod mechanism is in most cases negligible and may be disregarded.

Fig. 1

The torque produced on the shaft by the inertia forces of the reciprocating masses of the connecting-rod-and-crank mechanism of one cylinder is determined by the series⁵:

\[ M''=\sum M''_{h}\cdot \sin h\eta t, \tag{4} \]

where \(h=1, 2, 3\ldots\).

The coefficients \(M''_{h}\) (the amplitudes of the harmonic components of the torque on the shaft due to the inertia forces of the reciprocating masses) have the following values:

\[ \begin{aligned} M''_{1}&=\frac{G_{h}}{g}\,r^{2}\eta^{2} \left(\frac{\mu}{4}+\frac{\mu^{3}}{16}+\frac{15}{512}\mu^{5}+\ldots\right),\\ M''_{2}&=\frac{G_{h}}{g}\,r^{2}\eta^{2} \left(-\frac{1}{2}-\frac{\mu^{4}}{32}-\frac{\mu^{6}}{32}-\ldots\right),\\ M''_{3}&=\frac{G_{h}}{g}\,r^{2}\eta^{2} \left(-\frac{3}{4}\mu-\frac{9}{32}\mu^{3}-\frac{81}{512}\mu^{5}-\ldots\right),\\ M''_{4}&=\frac{G_{h}}{g}\,r^{2}\eta^{2} \left(-\frac{\mu^{2}}{4}-\frac{\mu^{4}}{8}-\frac{\mu^{6}}{16}-\ldots\right), \end{aligned} \tag{5} \]

and so on.

where \(G_{h}\) is the weight of the reciprocating masses of the connecting-rod-and-crank mechanism of one cylinder;

\(r\) is the crank radius;

\[ \mu=\frac{r}{l}, \]

where \(l\) is the length of the connecting rod;

\[ g=9.81\ \text{m/sec}^{2}. \]

The phase angles of the component harmonic moments in formula (4) are equal to \(0^\circ\) or \(180^\circ\), depending on the sign of \(M''_{h}\).

Separate harmonic analysis of the tangential forces due to gas pressure in the cylinder and due to the inertia forces of the reciprocating masses offers great convenience for calculating forced torsional vibrations of the shaft at various spee—

the growths of its rotation. In this case, with sufficient accuracy for practice, the indicator diagram, and consequently also the expansion of \(M^1\) into series (3), may be regarded as independent of the number of revolutions \(n\) of the engine (of course, only when \(n\) varies within comparatively narrow limits). The coefficients of series (4), however, must be calculated by formulas (5) for each \(n\) separately.

Summing series (3) and (4) for a given \(n\) gives the final expression for the total torque produced on the engine shaft by one cylinder:

\[ M = M_0 + \sum M_h \cdot \sin(h\eta t + \beta_h), \tag{6} \]

where \(M_0\) retains its previous value. The quantities \(M_h\) and \(\beta_h\) are determined by adding the corresponding (of the same order) harmonics \(M'_h\) and \(M''_h\), taking into account their phase angles. It is evident that to each number of revolutions of the engine shaft there correspond different values of the quantities \(M_h\), \(\beta_h\), and \(\eta\).

Each harmonic moment \(M_h\) entering into series (6) may be represented in the following form:

\[ M_h = C_h \cdot F \cdot r, \tag{7} \]

where \(F\) is the piston area in \(\mathrm{cm}^2\);

\(C_h\) is the harmonic coefficient, measured in \(\mathrm{kg}/\mathrm{cm}^2\).

The construction of the theoretical torsiogram and the determination of the actual degree of nonuniformity of rotation must be carried out separately for each specified operating speed of the engine. If this number of revolutions coincides with one of the critical speeds of rotation of the shaft, then one (the corresponding) harmonic of the torque enters into resonance with the natural vibrations of the shafting, thereby causing large amplitudes of forced vibrations. All other harmonics are then outside resonance, and the amplitudes of the vibrations caused by them are comparatively small⁶.

The methods for determining the amplitudes and phase angles of the forced vibrations produced by resonant and nonresonant harmonics are different and must be considered separately.

II. FORCED VIBRATIONS OUTSIDE RESONANCE

In calculating forced vibrations produced by nonresonant harmonics of the torque, one may neglect all damping forces (resistances) inevitably arising in the installation during torsional vibrations of its shaft. Investigations by Holzer, Porter⁷, and others have shown that these forces noticeably reduce the amplitudes of the forced vibrations of the masses of an elastic system only at resonance or in the immediate vicinity of it, i.e., in very narrow regions near the critical rotational speeds of the shaft. The influence of damping forces on the magnitude of the amplitudes of forced vibrations is completely negligible even at a small distance from resonance. This circumstance is explained by the small magnitude of the damping forces in comparison with the inertia forces of the oscillating masses.

In studying the laws of forced vibrations caused by nonresonant harmonics of the torque, it is expedient to make use of Rayleigh’s theorem on the reciprocity of forced vibrations in an elastic system with many degrees of freedom⁸. As applied to forced torsional vibrations, this theorem may be stated as follows: the amplitude $\varphi_{k,i}$ of the vibration of shaft section $k$ caused by a harmonic moment $M$ applied at shaft section $i$ is equal to the amplitude $\varphi_{i,k}$ of the vibration of shaft section $i$ obtained under the action of the same moment $M$ at section $k$, i.e.,

$$ \varphi_{k,i}=\varphi_{i,k}. $$

⁹

Let us consider elastic systems with one flywheel (Fig. 2) and with two flywheels (Fig. 3), assuming that the external exciting moment acts only on the extreme right-hand mass of the shafting¹⁰. According to Rayleigh’s theorem, an exciting moment $M$ acting on mass $i$ of the engine causes vibrations of the right-hand mass of the system (the flywheel or the rotor of the dynamo) with the same amplitu-

—that which mass \(i\) receives under the action of this moment \(M\) on the right-hand end of the shaft, i.e.

\[ \varphi_{\mathrm{I},i}=\varphi_{i,\mathrm{I}}. \tag{8} \]

For a system with one flywheel (see Fig. 2);

\[ \varphi_{\mathrm{II},i}=\varphi_{i,\mathrm{II}} \tag{9} \]

for a system with two flywheels (see Fig. 3), where \(\varphi_{\mathrm{I},i}\) is the amplitude of the forced oscillations of the flywheel caused by the moment \(M\) acting on mass \(i\) of the engine; \(\varphi_{i,\mathrm{I}}\) is the amplitude of the forced oscillations of mass \(i\) of the engine, produced by the same moment \(M\), but applied to the flywheel. The meaning of the indices in equality (9) is analogous.

Fig. 2

Fig. 3

To determine \(\varphi_{\mathrm{I},i}\) or \(\varphi_{\mathrm{II},i}\) from relations (8) and (9), it is first necessary to find the dependence between some moment of a given frequency applied to the flywheel (in general, to the extreme right-hand mass of the system) and the amplitudes of oscillation of the engine masses caused by it. This dependence is in fact established in the form of a relation between the amplitudes of oscillation of the masses of the installation and the residual moment \(R\) at the right-hand end of the shafting (for example, when calculating the natural frequencies by the Tolle method), by computing a series of tables for various values of the angular velocity of oscillation \(\omega\). These computations can be considerably shortened in the investigation of power installations with in-line engines having equal stiffnesses of the sections of the crankshaft between the axes of each two adjacent cylinders and equal moments of inertia of the reduced masses of each cylinder (this condition is satisfied in practice for all heavy and many light internal-combustion engines).

In installations with engines of this kind, the amplitudes of oscillation of the engine masses are determined by the following dependences¹¹, valid when calculating the system from left to right, i.e. in the assump—

assumption that the external exciting moment acts only on the extreme right-hand mass of the shafting:

\[ \left. \begin{aligned} \varphi_2&=\varphi_1(1-A),\\ \varphi_3&=\varphi_1(1-3A+A^2),\\ \varphi_4&=\varphi_1(1-6A+5A^2-A^3),\\ \varphi_5&=\varphi_1(1-10A+15A^2-7A^3+A^4),\\ \varphi_6&=\varphi_1(1-15A+35A^2-17A^3+8A^4-A^5),\\ &\text{and so on.} \end{aligned} \right\} \tag{10} \]

where \(A=\dfrac{\Theta\omega^2}{c}\) is the principal parameter of the vibration;

\(\Theta\) is the moment of inertia of the flywheel masses of each engine cylinder with respect to the shaft axis;

\(c\) is the stiffness of the sections of the crankshaft between the axes of each two adjacent cylinders;

\(\omega\) is the angular velocity of the vibration, i.e., the angular velocity of the external exciting moment applied to the extreme right-hand mass of the system.

For \(\varphi_1=1\) radian, expressions (10) become:¹⁹

\[ \left. \begin{aligned} \varphi_2&=1-A,\\ \varphi_3&=1-3A+A^2,\\ \varphi_4&=1-6A+5A^2-A^3,\\ \varphi_5&=1-10A+15A^2-7A^3+A^4,\\ \varphi_6&=1-15A+35A^2-17A^3+8A^4-A^5,\\ &\text{and so on.} \end{aligned} \right\} \tag{11} \]

These quantities \(\varphi\) may be called the relative amplitudes of vibration of the engine masses (i.e., referred to the amplitude of vibration of the first mass).

The numerical values of \(\varphi\), calculated from formulas (11) for various values of the parameter \(A\), are given in Table I (see the appendix).

The relative amplitude of vibration of the flywheel in the installation according to Fig. 2 is determined by the obvious expression:

\[ \varphi_I=\varphi_m-\frac{\displaystyle\sum_{i=1}^{i=m}\theta\omega^2\varphi_i}{c_{mI}}, \]

¹⁹ In formulas (10), (11), and below, the numbering of the engine masses begins from the left-hand end of the installation according to Figs. 2 and 3.

or

\[ \varphi_{\mathrm I}=\varphi_m-\frac{\theta\omega^2}{\beta_1 c}\sum_{i=1}^{i=m}\varphi_i =\varphi_m-\frac{A}{\beta_1}\sum_{i=1}^{i=m}\varphi_i, \tag{12} \]

where \(\varphi_m\) is the amplitude of oscillation of the last right-hand mass of the engine when \(\varphi_1=1\);

\(\varphi_i\) is the relative amplitude of oscillation of the \(i\)-th mass of the engine;

\(\varphi_{\mathrm I}\) is the relative amplitude of oscillation of the flywheel;

\(c_{m\mathrm I}\) is the stiffness of the shaft segment between the last cylinder of the engine and the flywheel;

\[ \beta_1=\frac{c_{m\mathrm I}}{c} \]

is the relative stiffness of this shaft segment;

\(m\) is the number of cylinders of the engine;

\[ \sum_{i=1}^{i=m}\varphi_i \]

is the algebraic sum of the relative amplitudes of oscillation of all the engine masses.

The values of

\[ \sum_{i=1}^{i=m}\varphi_i \]

as a function of \(A\) are given in Tables II and III (see the appendix) for four- and six-cylinder engines.¹² The relative amplitude of oscillation of the second flywheel, or of the dynamo rotor, in the installation according to Fig. 3 is determined by the expression:

\[ \varphi_{\mathrm{II}}=\varphi_{\mathrm I}- \frac{\displaystyle\sum_{i=1}^{i=m}\theta\omega^2\varphi_i+\theta_{\mathrm I}\omega^2\varphi_{\mathrm I}} {c_{\mathrm I,\mathrm{II}}}. \]

or

\[ \begin{aligned} \varphi_{\mathrm{II}} &=\varphi_{\mathrm I}-\frac{\theta\omega^2}{\beta_2 c} \left(\sum_{i=1}^{i=m}\varphi_i+\alpha_{\mathrm I}\varphi_{\mathrm I}\right) \\ &=\varphi_{\mathrm I}-\frac{A}{\beta_2} \left(\sum_{i=1}^{i=m}\varphi_i+\alpha_{\mathrm I}\varphi_{\mathrm I}\right). \end{aligned} \tag{13} \]

Here \(\varphi_{\mathrm{II}}\) is the relative amplitude of oscillation of the second flywheel, or of the dynamo rotor;

\(c_{\mathrm{I,II}}\) is the stiffness of the shaft section between the first and second flywheels;

\[ \beta_2=\frac{c_{\mathrm{I,II}}}{c} \]
is the relative stiffness of this shaft section;

\(\Theta_{\mathrm{I}}\) is the moment of inertia of the first flywheel;

\(\Theta_{\mathrm{II}}\) is the moment of inertia of the second flywheel (the dynamo rotor);

\[ \alpha_{\mathrm{I}}=\frac{\Theta_{\mathrm{I}}}{\Theta}. \]

The magnitude of the relative amplitude \(\varphi_{\mathrm{I}}\) of oscillation of the first flywheel, entering into formula (13), is determined by expression (12).

The residual moment \(R\), when calculating the system from left to right, as is known, represents the amplitude of that external exciting harmonic moment which must be applied to the extreme right-hand mass of the installation in order to balance the inertial moments of all the masses of the shafting, oscillating with angular velocity of oscillation \(\omega\) ²¹.

In an installation with one flywheel (see Fig. 2), the residual moment at the right end of the shaft is equal to the algebraic sum of the maximum inertial moments of all the masses of the engine and the flywheel:

\[ R=-\left(\sum_{i=1}^{i=m}\theta\omega^2\varphi_i+\theta_{\mathrm{I}}\omega^2\varphi_{\mathrm{I}}\right)= \]

\[ =-\theta\omega^2\left(\sum_{i=1}^{i=m}\varphi_i+\alpha_{\mathrm{I}}\varphi_{\mathrm{I}}\right) =-Ac\left(\sum_{i=1}^{i=m}\varphi_i+\alpha_{\mathrm{I}}\varphi_{\mathrm{I}}\right), \tag{14} \]

and, analogously, in an installation with two flywheels (see Fig. 3):

²¹ It is necessary to distinguish the angular velocity of oscillation \(\omega\) (sometimes called the “angular frequency,” Drehschnelle) from the angular velocity of the oscillatory motion of the mass. The former is related to the frequency of oscillation \(N\) by the expression

\[ \omega=\frac{\pi N}{30}\ \mathrm{sec}^{-1}, \]

where \(N\) is equal to the frequency of the exciting harmonic moment (the number of cycles of its variation per minute). The quantity \(\omega\) should be understood as the angular velocity of rotation of the oscillation vector \(\varphi_i\), equal to the angular velocity of rotation \(h\eta\) of the vector of the harmonic moment. In an established oscillatory process the quantity \(\omega\) is constant. The angular velocity of the oscillatory motion of the mass is equal to

\[ \frac{d\varphi_i}{dt} \]

and is variable in time.

\[ R=-\left(\sum_{i=1}^{i=m}\theta\omega^2\varphi_i+\theta_{\mathrm I}\omega^2\varphi_{\mathrm I} +\theta_{\mathrm{II}}\omega^2\varphi_{\mathrm{II}}\right) \]
\[ =-Ac\left(\sum_{i=1}^{i=m}\varphi_i+\alpha_{\mathrm I}\varphi_{\mathrm I} +\alpha_{\mathrm{II}}\varphi_{\mathrm{II}}\right). \tag{15} \]

By these formulas the numerical values of \(R\) can be calculated for the particular system under investigation at various \(\omega\). The quantities \(\varphi_{\mathrm I}\) and \(\varphi_{\mathrm{II}}\) entering here are determined, respectively, by expressions (12) and (13). The values of \(\sum_{i=1}^{i=m}\varphi_i\), as before, are taken directly from the appended tables.

The quantity \(R\) thus found is the amplitude of that external harmonic moment which must be applied to the extreme right-hand mass of the system (the flywheel or the rotor of the dynamo) in order to produce oscillations of the first (extreme left-hand) mass of the engine with amplitude \(\varphi_1=1\). The corresponding amplitudes of oscillation of all the other masses are given by formulas (11), (12), (13).

Formulas (10)—(13) show that, when an exciting harmonic moment acts only on the right-hand end of the elastic system, the amplitudes of oscillation of all the masses are proportional to the amplitude of oscillation of the first mass \(\varphi_1\). Equally, according to formulas (14) and (15), the quantity of the residual moment \(R\) is proportional to \(\varphi_1\), or conversely: the amplitude \(\varphi_1\), and consequently also the amplitudes of oscillation of all the masses of the installation, are proportional to the amplitude of the external exciting harmonic moment applied to the right-hand end of the shafting.

Thus, if a harmonic moment with amplitude \(M\), instead of \(R\) (but with the same angular velocity \(\omega\)), acts on the right-hand end of the system, then the amplitude of oscillation of the first mass of the engine caused by this moment is determined from the relation¹³

\[ \varphi_{1,\mathrm I\,abs}=1\cdot\frac{M}{R}, \]

and the amplitudes of all the other masses from the relation:

\[ \varphi_{i,\mathrm I\,abs}=\varphi_i\cdot\frac{M}{R}. \]

But on the basis of Rayleigh’s theorem, according to formula (8), the amplitude \(\varphi_{1,\mathrm I abs}\) is equal to the amplitude \(\Phi_{\mathrm I,1 abs}\) of the forced oscillations of the ma—

...of the flywheel obtained under the action of the moment \(M\) on engine mass \(i\), i.e.

\[ \varphi_{\mathrm{I},i\,abs}=\varphi_i\cdot \frac{M}{R}. \tag{16} \]

Here \(\varphi_i\) is the relative amplitude of oscillation of mass \(i\), determined from formulas (11).

Each of the harmonic exciting moments \(M_1, M_2 \ldots M_m\), having the same frequency and acting respectively on masses \(1, 2 \ldots m\) of the engine, causes forced oscillatory motions of the flywheel with amplitudes:

\[ \left. \begin{aligned} \varphi_{\mathrm{I},1\,abs} &= \varphi_1\cdot \frac{M_1}{R},\\ \varphi_{\mathrm{I},2\,abs} &= \varphi_2\cdot \frac{M_2}{R},\\ &\ldots\ldots\ldots\ldots\\ \varphi_{\mathrm{I},i\,abs} &= \varphi_i\cdot \frac{M_i}{R},\\ &\ldots\ldots\ldots\ldots\\ \varphi_{\mathrm{I},m\,abs} &= \varphi_m\cdot \frac{M_m}{R}. \end{aligned} \right\} \tag{17} \]

The complete, resultant amplitude of the forced oscillations of the flywheel under the simultaneous action of all the moments \(M_1, M_2 \ldots M_m\), applied to the corresponding engine masses, can now be found by geometric summation of the amplitudes of all component oscillations:

\[ \varphi_{\mathrm{I}\,abs} = \frac{\overrightarrow{M_1}}{R}\,\varphi_1 + \frac{\overrightarrow{M_2}}{R}\,\varphi_2 + \ldots + \frac{\overrightarrow{M_m}}{R}\,\varphi_m, \tag{18} \]

If:

\[ M_1=M_2=\ldots=M_m=M, \]

which is usually the case in practice when the diagrams of tangential forces are identical for all cylinders of the engine, then

\[ \varphi_{\mathrm{I}\,abs} = \frac{\overrightarrow{M}}{R}\sum \varphi_i \tag{19} \]

Here \(\overrightarrow{\sum \varphi_i}\) is the geometric sum of the relative amplitudes of oscillation of all the engine masses, or, more precisely, of all those of its masses on which the harmonic moments \(M\) of the given order act.

..., determined graphically¹⁴ (see below) with account taken of the phase angles of the component oscillations.

In the general case, the exciting moment

\[ M \cdot \sin (h\eta t+\beta_h), \]

of the \(h\)-th order, acting on the \(i\)-th mass of the elastic system, will cause the motion of this mass according to the law:

\[ (\varphi)=\varphi_i \cdot \sin (h\eta t+\beta_h-\varepsilon). \tag{20} \]

In a precisely analogous manner, according to the law:

\[ (\varphi_k)=\varphi_k \cdot \sin (h\eta t+\beta_h-\varepsilon), \tag{21} \]

all the other masses of the shafting will now also move. Here \(\varepsilon\) is the phase angle between the exciting moment and the forced oscillation. Geometrically, \(\varepsilon\) is the angle between the vector of the moment \(M\) and the vector of the oscillation \(\varphi_i\). In the absence of damping forces in a system oscillating in the non-resonant region, the phase angle \(\varepsilon\) is equal to \(0^\circ\) or \(180^\circ\), i.e., all the masses of the shafting have identical or directly opposite phases. Therefore, regarding \(M\) as essentially positive, expressions (20) and (21) for forced oscillations outside resonance may be written in the form:

\[ (\varphi_i)=\varphi_i \cdot \sin (h\eta t+\beta_h), \]

\[ (\varphi_k)=\varphi_k \cdot \sin (h\eta t+\beta_h), \]

where the positive values \(\varphi_i\) and \(\varphi_k\) correspond to the angle \(\varepsilon=0^\circ\), i.e., when the oscillation vectors coincide in direction with the vector of the moment \(M\), while negative values of \(\varphi_i\) and \(\varphi_k\) correspond to the phase angle \(\varepsilon=180^\circ\). In this case the vectors of the harmonic moment and of the forced oscillation of the corresponding mass are mutually opposite.

The phase angle \(\varepsilon\) between the exciting harmonic moment \(M\) and the oscillation of the point of its application is equal to \(0^\circ\) or \(180^\circ\) only in those cases when only one harmonic moment acts on the elastic system (only on one mass of the shafting), or when all moments of the given order, applied to the masses of the engine, have identical or directly opposite phases. This angle differs from \(0^\circ\) and \(180^\circ\) if the harmonic moments (of the order under investigation) acting on the masses of the engine differ in phase.

To determine the magnitude of \(\vec{\Sigma}\varphi_i\), entering into formula (19), and the phase angle of the resulting (total) oscillation of the flywheel, it is necessary to construct the so-called “phase” diagram of the exciting harmonic moments of the given order. The phase diagram represents the mutual arrangement of the vectors of the harmonics \(M\) acting on masses \(1, 2 \ldots m\) of the engine at each instant of time. The arrangement of the vectors \(M\) in this diagram is determined by the order of the harmonic exciting moments being investigated, by the type of engine (two- or four-stroke), by the number of its cylinders, and by the order in which ignition occurs in them.

In the phase diagram the moment vectors are placed one after another clockwise, in the order of ignition in the cylinders, the angle between every two neighboring vectors being equal to \(h\alpha\) degrees, where

\[ \alpha = \frac{360^\circ}{m} \quad — \tag{22} \]

in two-stroke engines;

\[ \alpha = \frac{720^\circ}{m} \quad — \tag{23} \]

in four-stroke engines;

\(m\) — the number of engine cylinders;
\(h\) — as before — the order of the harmonic.^24

In Fig. 4, by way of illustration, phase diagrams are shown for a four-stroke four-cylinder engine, with the ignition sequence in the cylinders \(1—2—4—3\).

In Fig. 4:
Shaft diagram.
Ignition sequence: \(1—2—4—3\).
Phase diagrams:
\(n\ \tfrac{1}{2},\ 2\tfrac{1}{2}\);
\(n\ 1,\ 3,\ 5\ldots\);
\(n\ 1\tfrac{1}{2},\ 3\tfrac{1}{2}\ldots\);
\(n\ 2,\ 4,\ 6\ldots\).

Fig. 4

^24 For examples of constructing phase diagrams, see N. Urvantsev, Critical Numbers of Revolutions in Diesel Installations, 1931.

The values of the relative amplitudes \(\varphi_i\) of the oscillations of the engine masses, calculated by formulas (11) or taken directly from Table I (see the Appendix), must now be laid off along the directions of the moment vectors in the phase diagram of the given order; in doing so, positive \(\varphi_i\) are laid off from the center of the diagram outward, and negative ones—from outside toward the center. The closing side of the polygon constructed on these vectors \(\varphi_i\) gives the required value \(\overrightarrow{\sum \varphi_i}\) (Fig. 5). The direction of this closing side determines the position of the vector of the resulting oscillation \(\varphi_{Iabc}\) of the flywheel, relative to the vectors of the exciting harmonic moments.

Fig. 5

Let us denote by \(\varepsilon_0\) (Fig. 5) the angle between the closing vector \(\overrightarrow{\sum \varphi_i}\) and, for example, the vector of the harmonic moment acting on the first crank of the crankshaft. The magnitude of this angle may be measured directly from the drawing. The true phase angle \(\gamma\) of the resultant oscillation of the flywheel relative to the first crank, i.e., the angle between the vector \(\varphi_{Iabc}\) and the axis of abscissas at the instant when the crank passes through the upper dead center, at \(t=0\), is evidently equal to the difference between the phase angle \(\beta_h\) of the harmonic of the exciting moment and the angle \(\varepsilon_0\):

\[ \gamma = \beta_h - \varepsilon_0 . \tag{24} \]

Now the forced oscillatory motion of the flywheel, created by harmonic moments of the \(h\)-th order acting on all the cranks of the crankshaft, will be represented by the equation:

\[ \varphi=\varphi_{\mathrm{I}\,abs}\cdot \sin(h\eta t+\gamma), \tag{25} \]

in which the numerical values of all the quantities entering into it are already known.

Similar calculations of \(\varphi_{\mathrm{I}\,abs}\) and the determination of \(\sum \varphi_i\) and \(\gamma\) from the phase diagram must be carried out for all, or at least for the most important harmonics of the torque—large ones and those close to resonance—at the given number of revolutions of the engine. The most significant here will be the harmonics of the principal orders, i.e., those orders for which all the moment vectors in the phase diagram coincide in direction.

According to what was said above, this coincidence of vectors will occur when

\[ h\alpha=h\cdot \frac{360^\circ}{m}=360^\circ \]

in two-stroke engines,

\[ h\alpha=h\cdot \frac{720^\circ}{m}=360^\circ \]

in four-stroke engines,

i.e., when

\[ h=m \quad \text{and} \quad h=\frac{m}{2} \]

respectively—in two- and four-stroke engines.

Thus, the principal orders of the torque harmonics are multiples of the number of firings in the engine cylinders during one revolution of its shaft.

III. FORCED OSCILLATIONS AT RESONANCE

All those rotational speeds of the engine shaft at which one of the natural frequencies of oscillation of the shafting coincides with the frequency of one of the torque harmonics on the shaft are its critical speeds.

The critical rotational speeds of the engine shaft are determined by the expression:

\[ \eta_{kp}=\frac{\omega_0}{h}, \tag{26} \]

where \(\omega_0\) is the angular velocity of the natural oscillations of the elastic system, connected with the natural frequency of oscillations \(N_0\) by the relation:

\[ \omega_0=\frac{\pi N_0}{30}\ \mathrm{sec}^{-1}; \]

\(h\) is the order of the resonating harmonic, successively taking the values \(1, 2, 3\ldots\) in two-stroke engines and \(1/2; 1; 1\frac12\ldots\) in four-stroke engines.

From the existence of an entire series of natural vibration frequencies of the multi-mass system and from the multiplicity of harmonic components of the torque there follows the existence of many critical speeds in every power plant with a multi-cylinder engine.

In forced vibrations of the shafting with angular velocity \(\omega_0\), i.e., at resonance, the residual moment \(R\), as is known, becomes zero. Now the application of formula (19) for determining the amplitudes of the forced vibrations of the flywheel, produced by the simultaneous action of resonating harmonics of order

\[ h=\frac{\omega_0}{\eta_{kp}} \]

on all crank throws of the crankshaft, gives

\[ \varphi_{I\,abs}=\infty, \]

which is quite natural for resonance without resistances. At the same time the vibration amplitudes of all the remaining masses of the installation also become infinitely large. In practice, however, the growth of vibration amplitudes is limited by damping forces inevitably arising during torsional vibrations of the shafting.

In the most general case, knowing the specific characteristics of the elastic reduced system, of the exciting moments, and of the damping forces, it would be possible to write the differential equations of motion for each mass of the installation under investigation. The joint solution of the resulting system of equations would have to give the equations of motion of all the masses and, consequently, the complete picture of the forced torsional vibrations of the shaft. But such an approach is almost not used in practice, since the solution of the indicated system of differential equations is extremely complex and, in general, is possible (in quadratures) only with the introduction of a number of simplifying assumptions concerning the nature of the damping forces, etc. This path is sufficiently convenient only in the investigation of simple two- or three-mass systems.

For the analysis of resonant torsional vibrations of a shaft with a large number of masses, as a rule, the energy method proposed by Wydler (Wydler)\(^{25}\) is used, based on the following:

The growth of the amplitudes of forced vibrations of the shafting when the engine operates at one of its critical speeds will cease only when the work of the harmonic exciting moments acting on the shaft, performed by them during each—

\(^{25}\) H. Wydler, Drehschwingungen in Kolbenmaschinenanlagen, 1922.

each cycle of oscillation, will be equal to the work absorbed during this cycle by the damping forces.

WORK OF THE EXCITING MOMENTS

Let us suppose that on mass $i$ of an elastic system, oscillating according to the law: ^26

$$ (\varphi_i)=\varphi_{i\,abs}\cdot \sin((h\eta t+\beta_h-\varepsilon), \tag{27} $$

there acts an exciting moment:

$$ (M_i)=M_i\cdot \sin(h\eta t+\beta_h). \tag{28} $$

For each cycle of oscillation of mass $i$ this moment performs the work:

$$ A_{excit}=\int_{t=0}^{T}(M_i)\cdot d\varphi_{i\,abs} =\pi M_i\cdot \varphi_{i\,abs}\cdot \sin\varepsilon, \tag{29} $$

where

$$ T=\frac{2\pi}{h\eta} \tag{30} $$

is the period of oscillation.

If there are no other external harmonic moments in the installation, except $M_i$, applied to mass $i$, then the phase angle $\varepsilon$ in expressions (27) and (29) depends exclusively on the magnitude and character of the damping forces and on the ratio between $h\eta$ and $\omega_0$. For oscillations outside resonance $(h\eta\ne\omega_0)$:

$$ 0<\varepsilon<180^\circ, $$

and at resonance $(h\eta=\omega_0)$:

$$ \varepsilon=90^\circ, $$

i.e., the forced resonant oscillations of mass $i$ lag by $90^\circ$ behind the moment $M_i$ acting on it. To this value of the phase angle there corresponds the greatest possible work of the exciting moment during a cycle of oscillation:

$$ A_{excit}=\pi M_i\cdot \varphi_{i\,abs}. \tag{31} $$

But the value $\varepsilon=90^\circ$ between the moment and the oscillation of the point of its application is maintained at resonance only in the case where only one exciting moment acts on the system—

^26 The quantities $(\varphi_i)$ and $(M_i)$ denote, respectively, the instantaneous values of the angle of deflection of the mass and of the moment acting on it.

moment, at one of its points. The phase angle \(\varepsilon\) can turn out to be quite different in resonant oscillations of a many-mass system when several harmonic moments of the same frequency, but with different phases, act on it at its different points.¹⁵ Here the oscillatory motion of each mass is the result of the superposition of its individual component oscillations, produced by each of the exciting harmonic moments applied to the system.

If the masses \(1,2\ldots m\) of the engine, oscillating according to the law:

\[ \left. \begin{aligned} (\varphi_1)&=\varphi_{1\,abs}\cdot \sin(h\eta t+\beta_{h,1}-\varepsilon_1),\\ (\varphi_2)&=\varphi_{2\,abs}\cdot \sin(h\eta t+\beta_{h,2}-\varepsilon_2),\\ &\ldots\ldots\ldots\ldots\ldots\\ (\varphi_i)&=\varphi_{i\,abs}\cdot \sin(h\eta t+\beta_{h,i}-\varepsilon_i),\\ &\ldots\ldots\ldots\ldots\ldots\\ (\varphi_m)&=\varphi_{m\,abs}\cdot \sin(h\eta t+\beta_{h,m}-\varepsilon_m), \end{aligned} \right\} \tag{32} \]

are acted upon respectively by the exciting harmonic moments:

\[ \left. \begin{aligned} (M_1)&=M_1\cdot \sin(h\eta t+\beta_{h,1})\\ (M_2)&=M_2\cdot \sin(h\eta t+\beta_{h,2})\\ &\ldots\ldots\ldots\ldots\\ M_i)&=M_i\cdot \sin(h\eta t+\beta_{h,i}),\\ &\ldots\ldots\ldots\ldots\\ (M_m)&=M_m\cdot \sin(h\eta t+\beta_{h,m}), \end{aligned} \right\} \tag{33} \]

then the total work performed by all these moments during one cycle of oscillation will be determined, according to formula (29), by the expression:

\[ A_{excit}=\sum \pi M\cdot \varphi_{i\,abs}\cdot \sin\varepsilon_i. \tag{34} \]

In the presence of damping forces in the installation, the phase angles \((\beta_{h,i}-\varepsilon_i)\) in expressions (32) are different for each of the masses of the shafting. Mutual inequality of \((\beta_{h,1}-\varepsilon_1)\), \((\beta_{h,2}-\varepsilon_2)\), etc., generally occurs in any forced torsional oscillations, both at resonance and outside it. The absence of synchronism in the motion of the masses greatly complicates the investigation of forced torsional oscillations with allowance for damping forces, since it requires determining the phase angle \((\beta_{h,i}-\varepsilon_i)\) for each of

masses separately. But the solution of the problem can be simplified when analyzing only resonant forced vibrations.

Widler¹⁶ showed that at resonance, at least so long as the damping forces are not very large, the curve of the form of the forced vibrations of the shafting practically does not differ from the curve of the form of the free vibrations without resistances.

Thus, in analyzing forced resonant vibrations, one may, with accuracy quite sufficient for practice, assume the synchronism of the vibratory motion of all the masses of the system. This circumstance makes it possible conveniently to use equation (34) for calculations of resonant amplitudes.

Assuming now in expressions (32) equality of the absolute phase angles of vibration of the masses¹⁷:

\[ (\beta_{h,1}-\varepsilon_1)=(\beta_{h,2}-\varepsilon_2)=\ldots\ldots=(\beta_{h,i}-\varepsilon_i)=\gamma, \tag{36} \]

formula (35) may be rewritten in the form:

\[ A_{\mathrm{exc}}=\pi \sum M_i \cdot \varphi_{i\,\mathrm{abs}}\cdot \sin(\beta_{h,i}-\gamma), \]

or, for the same magnitude of the amplitudes of all harmonic moments of the given order acting on the crankshaft:

\[ A_{\mathrm{exc}}=\pi M \sum \varphi_{i\,\mathrm{abs}}\cdot \sin(\beta_{h,i}-\gamma). \tag{37} \]

The equality (36) adopted by us shows that the vectors of the forced vibrations of all the masses, under the action of harmonic moments of one order on all the crankpins of the engine crankshaft, are collinear at every instant of time. At the initial instant of time, when \(t=0\), all these vectors, rotating with angular velocity \(h\eta=\omega_0\), make the angle \(\gamma\) with the axis of abscissae.

Each of the terms:

\[ \varphi_{i\,\mathrm{abs}}\cdot \sin(\beta_{h,i}-\gamma), \]

entering under the summation sign in expression (37), may be regarded, according to Fig. 6, as the projection of the vector \(OB=\varphi_{i\,\mathrm{abs}}\), laid off in the direction of the vector of the harmonic moment \(M_i=OA\), acting on this mass, onto the line \(OII\), perpendicular to the line \(OI\), which determines the actual direction of the vector \(\varphi_{i\,\mathrm{abs}}\), i.e.

\[ OD=BC=\varphi_{i\,\mathrm{abs}}\cdot \sin(\beta_{h,i}-\gamma). \]

It follows from this that the quantity

\[ \sum \varphi_i{}_{\mathrm{abs}}\cdot \sin(\beta_{h,i}-\gamma) \]

in formula (37) for the total work of the exciting moments is the projection of the resultant vector \(\overrightarrow{\sum \varphi_i{}_{\mathrm{abs}}}\), constructed in the phase diagram of the exciting moments of the given order, onto the line perpendicular to the actual direction of the collinear vectors of vibration of all the masses. This resultant vector is determined as the closing side of the polygon constructed on the vectors of the amplitudes \(\varphi_i{}_{\mathrm{abs}}\) of vibration of all the engine masses, laid off in the directions of the corresponding vectors of the exciting moments in their phase diagram (Fig. 7):

Fig. 6

Fig. 7

\[ \overrightarrow{OE}=\sum \varphi_{i\,abs}, \tag{38} \]

\[ EF=\sum \varphi_{i\,abs}\cdot \sin(\beta_{h,i}-\gamma). \tag{39} \]

In equation (37), the value of the absolute phase angle \(\gamma\) of the oscillation of the masses remains as yet unknown, and, consequently, so do the values of the relative phase angles \(\beta_{h,i}-\gamma=\varepsilon_i\) between the exciting moments and the oscillations of the points at which they are applied.

It is evident that full resonance will occur under those conditions and relations which will ensure the greatest effectiveness of the action of the exciting moments, i.e., the greatest value of the work \(A_{excit}\) performed by them during each cycle of oscillation. According to equations (37), (39), and Fig. 7, the work \(A_{excit}\) is proportional to the segment \(EF\), i.e., to the projection of the vector

\[ \overrightarrow{OE}=\sum \varphi_{i\,abs} \]

onto the line \(OII\), perpendicular to the direction of the oscillation vectors (line \(OI\)). The segment \(EF\) reaches its maximum value:

\[ (EF)_{\max}=OE=\sum \varphi_{i\,abs} \]

when the angle between \(OE\) and \(OI\) is \(90^\circ\), i.e., at full resonance the vectors of the forced oscillations of all masses are arranged along the line \(OIII\), perpendicular to \(OE\), and lag by \(90^\circ\) behind the vector

\[ \sum \varphi_{i\,abs}. \]

The absolute phase angle of these oscillations \(\gamma_0\) is equal to the angle between the line \(OIII\) and the axis of abscissas (see Fig. 7). The value \(\gamma_0\) can be measured directly from the drawing.

In the case of full resonance, the work of the exciting moments is now determined by the formula:

\[ A_{excit}=\pi M\cdot \sum \varphi_{i\,abs}, \tag{40} \]

or

\[ A_{excit}=\pi M\cdot \varphi_{1\,abs}\cdot \sum \frac{\varphi_{i\,abs}}{\varphi_{1\,abs}} = \pi M\cdot \varphi_{1\,abs}\sum \varphi_i, \tag{41} \]

where

\[ \frac{\varphi_{i\,abs}}{\varphi_{1\,abs}}=\varphi_i \]

is the relative amplitude of oscillation of the \(i\)-th mass.

Formula (41), which is fundamental in calculating the resonance amplitudes of torsional oscillations, shows that \(A_{excit}\) is directly proportional to the amplitude of oscillation of the first mass of the engine.

In accordance with the above-mentioned Widler proposition concerning the practically exact similarity of the curves of the form of free and forced resonant vibrations, the quantities \(\varphi_i\) entering into formula (41) may be calculated by formulas (11) or taken from Table 1 (see the Appendix) for

\[ A=\frac{\theta\omega_0^2}{c}, \]

where \(\omega_0\) is the angular velocity of the natural vibrations of the system.

For an engine with a nonuniform distribution (along the length of the crankshaft) of the reduced masses of the connecting-rod-and-crank mechanisms, the quantities \(\varphi_i\) are computed, for example, by Tolle’s method.

The phase diagrams of the exciting moments, necessary for determining \(\sum \varphi_i\), are constructed according to the instructions in Section II of the present article. For harmonic moments of the principal orders, owing to the collinearity of their vectors, the geometrical sum of \(\varphi_i\) becomes an algebraic sum.

WORK OF THE DAMPING FORCES AND RESONANT AMPLITUDES OF VIBRATION

The method of calculating the work of the damping forces depends on the character of the most important resistances taken into account in the calculation and on the form of the analytical expression adopted for them.

Widler’s theory. Widler proposed^30 to consider all damping forces in the engine, without entering into their detailed analysis, as proportional to the first power of the velocity of the vibratory motion of the masses of the connecting-rod-and-crank mechanisms reduced to the crank radius. Widler’s theory is the simplest from the formal and computational standpoint, and finds a certain justification in the usual estimate of the character of damping forces adopted in the general theory of vibrations.

According to Widler, on the mass \(i\) of the engine, vibrating according to the law:

\[ (\varphi_i)=\varphi_{i\,\text{abs}}\cdot \sin(\omega t+\gamma), \]

there acts a damping force:

\[ P=-k_1 r\cdot \frac{d\varphi_i}{dt}, \]

where \(k_1\) is the resistance force (damping force) (in kg) encountered by the mass when it moves with a linear velocity of \(1\ \text{cm/sec}\);

\(r\) is the radius of reduction of the mass, equal to the crank radius.

The work absorbed by this damping force \(P\) during a cycle of vibration:

^30 See his work cited in footnote 25.

\[ A_r=\int_{t=0}^{T} P\cdot r\cdot d\varphi_i =\pi r^2 k_1\cdot \varphi_{i\,abs}^{\,2}\cdot \omega, \tag{42} \]

where

\[ T=\frac{2\pi}{\omega}. \]

The total work of the damping forces in the entire engine, i.e. in \(m\) cylinders during one cycle of oscillation:

\[ A_r=\sum \pi r^2 k_1\cdot \varphi_{i\,abs}^{\,2}\cdot \omega, \]

or, as before, introducing the relative amplitudes \(\varphi_i\) of the oscillation of the engine masses:

\[ A_r=\pi r^2 k_1\omega\cdot \varphi_{1\,abs}^{\,2}\cdot \sum \varphi_i^2. \tag{43} \]

Thus, the work absorbed by the velocity damping forces¹⁸ in the engine during one cycle of oscillation increases in proportion to the square of the amplitude of oscillation of the first mass, i.e. considerably faster than the work of the exciting moments, determined by formula (41).

Equating the right-hand sides of formulas (41) and (43) for \(A_{excit}\) and \(A^r\),

\[ \pi M\cdot \varphi_{1\,abs}\cdot \sum \varphi_i = \pi r^2\cdot k_1\cdot \omega\cdot \varphi_{1\,abs}^{\,2} \sum \varphi_i^2, \tag{44} \]

we obtain from this an expression for the maximum value of the amplitude of the forced resonant oscillations of the first engine mass, produced by the resonating harmonics \(M\):

\[ \varphi_{1\,abs} = \frac{M\sum \varphi_i}{r^2 k_1\omega \sum \varphi_i^2}. \tag{45} \]

It is evident that in Fig. 8 \(\varphi_{1\,abs}\) will be determined as the abscissa of the point of intersection of the curve \(A_r\) with the straight line \(A_{excit}\), since \(\varphi_{1\,abs}\) according to formula (45) corresponds to the established resonant oscillatory process.

Denoting:

\[ k_1=k\cdot F, \tag{46} \]

where \(F\) is the piston area (in \(\text{cm}^2\)) and expressing \(M\) through the harmonic coefficient \(C_h\) according to formula (7), the final expression for \(\varphi_{1\,abs}\) may be written in the following form:

\[ \varphi_{1\,abs}=\frac{C_h\cdot \sum \varphi_i}{kr\omega\cdot \sum \varphi_i^2}. \tag{47} \]

Here \(\sum^2\varphi_i\) is the arithmetic sum of the squares of the relative amplitudes of all the masses of the engine, i.e., only those masses of the installation upon which velocity-dependent damping forces are assumed to act. In ship installations, along with the damping forces in the engine, it is necessary to take into account the additional resistances arising during the oscillatory motion of the propeller. In this case, to the right-hand side of equation (44) one should add the work absorbed by the damping forces on the propeller¹⁹. In accordance with this, formula (47) will, of course, also change. In an analogous manner the velocity-dependent resistances in dynamo machines, etc., may be taken into account.

Fig. 8

The coefficient \(k\), entering into formula (47) and determined, according to formula (46), by the expression

\[ k=\frac{k_1}{F}\ \text{kg sec}/\text{cm}^3, \tag{48} \]

is the resistance force (in kg) encountered by the oscillating mass of the connecting-rod–crank mechanism of an engine cylinder, reduced to the crank radius, during its oscillatory motion with a velocity of \(1\ \text{cm}/\text{sec}\), referred to \(1\ \text{cm}^2\) of piston area.

On the basis of the experiments of a number of investigators, the coefficient \(k\) has the following values:

for heavy (oil) engines²⁰:

\[ k=0.005\text{--}0.006\ \text{kg sec}/\text{cm}^3; \]

for automobile engines:^34

\[ k = 0.0025\ \text{kg sec}/\text{cm}^3; \]

for aircraft engines:^35

\[ k = 0.001—0.002\ \text{kg sec}/\text{cm}^3; \]

for the same aircraft engines Manza^36 obtained:

\[ k = 0.00035—0.00055\ \text{kg sec}/\text{cm}^3. \]

Formula (45) gives the actual amplitude of the total forced vibrations of the first mass of the engine, produced by the simultaneous action of all resonating harmonics of the given order, on all, respectively, crankpins of the crankshaft.

To determine the absolute amplitudes of the resonant vibrations of all the other masses of the installation, it is necessary to multiply their relative amplitudes \(\varphi_i\) by \(\varphi_{\mathrm{I}\,\text{abs}}\). The quantities \(\varphi_i\) are calculated by formulas (11). The relative amplitudes of vibration of the flywheels may be found as follows.

For free vibrations of the engine—flywheel system (see Fig. 2):

\[ \sum_{1}^{m}\theta \omega_0^2 \varphi_i + \theta_{\mathrm{I}}\omega_0^2 \cdot \varphi_{\mathrm{I}} = 0, \]

whence

\[ \varphi_{\mathrm{I}} = - \frac{1}{\alpha_{\mathrm{I}}}\sum_{1}^{m}\varphi_i, \tag{49} \]

where

\[ \alpha_{\mathrm{I}} = \frac{\theta_{\mathrm{I}}}{\theta}; \]

\[ \sum_{1}^{m}\varphi_i \]

is the algebraic sum of the relative amplitudes of all masses of the engine.

^34 I. Plünzke, Drehschwingungen des Automobilmotors, Motorwagen, 1926.

^35 A. Stieglitz, Drehschwingungen in Reihenmotoren, Luftfahrtforschung, p. 133, 1929; there is a Russian translation of this article in the collection of translated works Torsional Vibrations of Crankshafts, published by the Air Force Academy of the Workers’ and Peasants’ Red Army, 1935.

^36 I. Manza, Die Bestimmung der Dämpfung von Drehschwingungen einer Flugmotorenkurbelwelle. Diss. T. H. Karlsruhe, 1933.

Similarly, in the engine—flywheel—dynamo installation (see Fig. 3):

\[ \sum_{1}^{m}\theta\omega_{0}^{2}\varphi_i+\theta_{\mathrm{I}}\omega_{0}^{2}\varphi_{\mathrm{I}}+\theta_{\mathrm{II}}\omega_{0}^{2}\varphi_{\mathrm{II}}=0, \]

or

\[ \varphi_{\mathrm{II}}= -\frac{\displaystyle\sum_{1}^{m}\varphi_i+a_{\mathrm{I}}\varphi_{\mathrm{I}}}{a_{\mathrm{II}}}, \tag{50} \]

where

\[ a_{\mathrm{II}}=\frac{\theta_{\mathrm{II}}}{\theta}. \]

The value of \(\varphi_{\mathrm{I}}\) entering formula (50) is determined by formula (12).

From the relative amplitudes \(\varphi_i\), \(\varphi_{\mathrm{I}}\), \(\varphi_{\mathrm{II}}\), determined by formulas (11), (49), (12), and (50), one can construct the general curve of the form of the free vibrations of the reduced system under investigation (Fig. 9).

Fig. 9

Now the absolute values of the amplitudes of the forced resonant vibrations of the flywheel or of the dynamo rotor:

\[ \varphi_{\mathrm{I}\,abc}=\varphi_{\mathrm{I}}\cdot \varphi_{1\,abc} =-\frac{1}{a_{\mathrm{I}}}\,\varphi_{1\,abc}\cdot \sum_{1}^{m}\varphi_i, \tag{51} \]

\[ \varphi_{\mathrm{II}\,abc}=\varphi_{\mathrm{II}}\cdot \varphi_{1\,abc} =-\frac{1}{a_{\mathrm{II}}}\,\varphi_{1\,abc}\cdot \left(\sum_{1}^{m}\varphi_i+a_{\mathrm{I}}\varphi_{\mathrm{I}}\right), \tag{52} \]

and the equations of their motion:

\[ (\varphi_{\mathrm{I}\,abc})=\varphi_{\mathrm{I}\,abc}\cdot \sin(\omega_0 t+\gamma_0), \tag{53} \]

\[ (\varphi_{\mathrm{II}})=\varphi_{\mathrm{II}\,abc}\cdot \sin(\omega_0 t+\gamma_0), \tag{54} \]

where the phase angle \(\gamma_0\) is found from the construction in Fig. 7.

Shannon’s studies (F. Shannon)\(^{37}\). Shannon, on the basis of experiments with an automobile engine running idle and driven by an electric motor, and also on the basis of processing data from other investigators, introduces the concept of the dimensionless damping coefficient (non-dimensional damping factor):

\[ \frac{\theta\omega_0}{k_0}, \]

where \(k_0\) is the damping coefficient in kg·cm per \(1\ \mathrm{sec}^{-1}\) of angular velocity of the vibrational motion (vibrational velocity) of the mass;

\(\theta\) is the moment of inertia of the reduced mass of the connecting-rod–crank mechanism of one cylinder;

\(\omega_0\) is the angular velocity of the natural oscillations of the shaft system.

The coefficient \(k_0\), for example, for the simplest single-mass system (Fig. 10) enters into the basic equation of forced oscillations:

\[ \theta\frac{d^2\varphi}{dt^2}+k_0\frac{d\varphi}{dt}+c\varphi=M\cdot \sin \omega t, \]

i.e. \(k_0\) is the moment of the damping forces in the engine cylinder, acting on the reduced mass of the cylinder mechanism when the velocity of its vibrational motion is \(1\ \mathrm{sec}^{-1}\).

According to Shannon’s experiments, the magnitude of the dimensionless damping coefficient

\[ \frac{\theta\omega_0}{k_0} \]

depends on the character of the curve of the form of the natural—

\(^{37}\) F. Shannon, Damping influences in torsional oscillation, Engineering, 1935, pp. 675, 702.

free vibrations of the shafting. A factor characterizing this curve may be the mean relative amplitude:

Fig. 10

\[ \varphi_{cp}=\frac{\displaystyle\sum_{1}^{m}\varphi_i}{m} \tag{55} \]

of the free vibrations of the engine masses, where \(m\) is the number of cylinders of the engine, and

\[ \sum_{1}^{m}\varphi_i \]

is the algebraic sum of the relative amplitudes.

The quantity \(\varphi_{cp}\), depending on the particular properties of the elastic vibrating system, may have any value in the range from 0 to 1:

when the node of vibration is located far outside the engine:

\[ \varphi_{cp}=\frac{\displaystyle\sum_{1}^{m}\varphi_i}{m}\cong 1; \]

but if the node of vibration is located exactly at the middle of the engine, then:

\[ \varphi_{cp}=\frac{\displaystyle\sum_{1}^{m}\varphi_i}{m}=0 \]

In the first case all the masses of the engine vibrate together, as a single whole; in the second, the direction of the vibratory motion of one half of the engine masses at every instant of time is opposite to the direction of motion of the other half.

The graphical relation between \(\frac{\theta\omega_0}{k_0}\) and \(\frac{\displaystyle\sum_{1}^{m}\varphi_i}{m}\), obtained by Pannon (Fig. 11), may be represented by the following equation:

\[ \frac{\theta\omega_0}{k_0} = 55-45\,\frac{\displaystyle\sum_{1}^{m}\varphi_i}{m}. \]

Hence the damping coefficient

36.

\[ k_0=\frac{\theta\omega_0}{55-45\,\dfrac{\displaystyle\sum_{1}^{m}\varphi_i}{m}} \tag{56} \]

The coefficients \(k_0\) according to Shannon and \(k\) according to Widler are, evidently, closely related to one another, since both characterize the damping forces in the engine that are proportional to the velocity of the oscillatory motion.

Fig. 11

In the figure: ordinate \(\dfrac{\theta\omega_0}{k_0}\); abscissa \(\dfrac{\sum \varphi_i}{m}\).
Left diagram: “form of a free vibration at high \(\omega\).”
Right diagram: “form of a free vibration at low \(\omega\).”

The moment of the damping forces acting on the reduced mass of one cylinder, according to Shannon:

\[ M_r=k_0\frac{d\varphi}{dt}. \]

According to Widler:

\[ M_r=kFr^2\cdot\frac{d\varphi}{dt}. \]

From the necessary equality of both these moments we have:

\[ k=\frac{k_0}{Fr^2}, \tag{57} \]

or, on the basis of formula (56), finally:

\[ k=\frac{\theta\omega_0}{F r^2}\cdot \frac{1}{\dfrac{\displaystyle\sum_{1}^{m}\varphi_i}{\displaystyle 55-45\frac{1}{m}}}. \tag{58} \]

The ratio \(\dfrac{\theta}{r^2}=m_{red}\) is the mass of the mechanism of one cylinder, reduced to the crank radius; consequently,

\[ k=m_{red}\cdot\omega_0\cdot \frac{1}{\dfrac{\displaystyle\sum_{1}^{m}\varphi_i}{\displaystyle 55-45\frac{1}{m}}}, \tag{59} \]

i.e., the damping coefficient \(k\) according to Widler, in accordance with Shannon’s experiments, depends on the reduced mass \(m_{red}\) of the cylinder mechanism, referred to the piston area \(F\), on the angular velocity of the natural oscillations of the system, and on the mean relative amplitude of oscillation of the engine masses.

Shannon estimates the degree of accuracy of his relation (see Fig. 11) at \(\pm 15\%\).

Lewis’s theory (F. Lewis)\(^{38}\). According to Lewis, the most important cause of the damping of forced torsional vibrations should be considered to be elastic hysteresis in the shaft.

Elastic hysteresis is characterized by the loss of energy in the metal during a cyclic change of the stress in it below the elastic limit. This absorption of energy is due to the imperfect elasticity of the metal. As a result of the deviation of every metal from Hooke’s law, the curve of strains and stresses in a rod subjected, for example, to alternating twisting, becomes a closed loop\(^{39}\). The area of this hyste-

\(^{38}\) See his work cited in footnote 32.

\(^{39}\) A rather extensive literature is devoted to the study of elastic hysteresis in metal; see, for example:

W. Thomson (Kelvin), “Proc. of Roy. Soc. of London”, Vol. 14 (1865), p. 289.
M. Guye, “Journal de Physique”, 1912, p. 620.
B. Hopkinson and F. Williams, “Proc. of Roy. Soc. of London”, Vol. 87 (1912), p. 502.
O. Föppl, “Z. d. V. D. I.”, 1928, S. 1293.
H. Kortum, “Z. d. V. D. I.”, 1930, S. 1426.
P. Ludwik u. R. Scheu, “Z. d. V. D. I.”, 1932, S. 683.
A. Appenrodt, “Z. d. V. D. I.”, 1936, S. 517.
O. Föppl, Becker, Heidekampf, Long-duration testing of materials. ONTI, 1935.
O. Föppl, Grundzüge der technischen Schwingungslehre, Berlin, 1931.

the hysteresis loop determines the energy absorbed by the metal during a cycle of change of load.

To calculate the hysteresis loss in the shaft of a power plant during its torsional vibrations, Lewis uses experimental data obtained by Rowett (F. Rowett).²¹

Rowett determined the hysteresis loss under torsional vibrations of thin-walled drawn tubes made of plain carbon steel having the composition: C—0.17%; Mn—0.24%; S—0.02%; P—traces, and a tensile strength (in the annealed state)

\[ \sigma_B = 22.4\ \text{t}/\text{dm}^2 . \]

By processing Rowett’s data, Lewis obtained the following expression for the energy \(\Delta A_h\), absorbed by a unit volume of the metal of a thin-walled tube during one cycle of vibration:

\[ \Delta A_h = 1.37 \cdot 10^{-10}\cdot \tau_{\max}^{2.3}\ \text{ft}\cdot\text{lb}/\text{dm}^3, \tag{60} \]

where \(\tau_{\max}\) is the maximum torsional stress in the tube in \(\text{lb}/\text{dm}^2\). Conversion of this formula into metric measures gives:

\[ \Delta A_h = 43.3 \cdot 10^{-10}\cdot \tau^{2.3}\ \text{kg cm}/\text{cm}^3, \tag{61} \]

where \(\tau_{\max}\) is in \(\text{kg}/\text{cm}^2\).

Formulas (60) and (61) are valid only for

\[ \tau_{\max} < 8000\ \text{lb}/\text{dm}^2\ (562\ \text{kg}/\text{cm}^2). \]

At higher stresses \(\tau_{\max}\), the hysteresis loss increases considerably faster than is established by the exponent 2.3.

When calculating the hysteresis loss in a solid shaft or in a hollow one but with a large wall thickness, it is necessary to take into account the variability of \(\tau_{\max}\) over its cross-section.

Let us isolate from a thick-walled hollow shaft with inner diameter \(d_i\) and outer diameter \(d_a\), of length \(l\), a coaxial cylinder of radius \(\rho\) and wall thickness \(d\rho\). The torsional stress (the maximum during the cycle of vibration) in the cross-section of this cylinder is:

\[ \tau'_{\max} = \tau_{\max}\cdot \frac{2\rho}{d_a}, \]

where \(\tau_{\max}\) is the maximum stress, during the cycle of vibration, in the outermost fibers of the shaft. The hysteresis loss in the isolated elementary cylinder:

\[ dA_h=43.3\cdot 10^{-10}\cdot(\tau'_{\max})^{2.3}\cdot dV = \]

\[ =2\pi\rho\cdot d\rho\cdot l\cdot 43.3\cdot 10^{-10}\cdot \left(\tau_{\max}\cdot \frac{2\rho}{d_a}\right)^{2.3}, \]

and the total hysteresis loss in the entire shaft over one cycle of oscillation:

\[ A_h=\int_{d_i/2}^{d_a/2} 2\pi\rho\cdot d\rho\cdot l\cdot 43.3\cdot 10^{-10}\cdot \left(\tau_{\max}\cdot \frac{2\rho}{d_a}\right)^{2.3} = \]

\[ =15.85\cdot 10^{-10}\cdot \tau_{\max}^{2.3}\cdot (1-\psi^{4.3})\cdot d_a^2\cdot l\ \text{kg cm}, \tag{62} \]

where

\[ \psi=\frac{d_i}{d_a}. \tag{63} \]

If we denote

\[ k_h=15.85\cdot \tau_{\max}^{2.3}\cdot(1-\psi^{4.3})\ \text{kg cm/cm}^3, \tag{64} \]

then formula (62) may be written in the form:

\[ A_h=k_h\cdot d_a^2\cdot l. \tag{65} \]

The hysteresis loss in the shaft of the entire installation:

\[ A_h=\sum k_h\cdot d_a^2 l, \tag{66} \]

where \(k_h\), \(d_a\), and \(l\) are different for each of the sections of the shaft. The hysteresis loss in the crankshaft of the engine, owing to the absence of the necessary experimental data, has to be taken as equal to the loss in a smooth equivalent shaft having the same outside diameter as the main journals of the crank.

The maximum torsional stress \(\tau_{\max}\) in formulas (62), (64), (66) is determined by the usual formula, separately for each cylindrical section of the shaft of the installation:

\[ \tau_{\max}=\frac{M_{\max}}{W_p}= \frac{16M_{\max}}{\pi d_a^3(1-\psi^4)}, \tag{67} \]

where \(W_p\) is the polar section modulus of the shaft cross-section.

The maximum moment \(M_{\max}\), arising due to torsional oscillations and twisting the shaft in the section between masses \(i\) and \(i+1\):

\[ M_{\max}=(\varphi_{i+1\ \mathrm{abs}}-\varphi_{i\ \mathrm{abs}})\cdot c_{i,i+1}, \tag{68} \]

where \(c_{i,i+1}\) is the stiffness of the shaft section between these masses.

Now:

\[ \tau_{\max}=\frac{\left(\varphi_{i+1\,abs}-\varphi_{i\,abs}\right)\cdot c_{i,i+1}}{W_p}, \]

and the total hysteresis loss in the shaft of the installation over a cycle of forced oscillations:

\[ A_h=15.85\cdot 10^{-10}\cdot \sum\left[ \frac{\left(\varphi_{i+1\,abs}-\varphi_{i\,abs}\right)\cdot c_{i,i+1}}{W_p} \right]^{2.3} \times \]

\[ \times\left(1-\psi^{4.3}\right)\cdot d_a^2\cdot l, \tag{69} \]

or, if the relative amplitudes of oscillation \(\varphi_i\) are introduced:

\[ A_h=15.85\cdot 10^{-10}\cdot \varphi_{1\,abs}^{\,2.3}\cdot \sum\left[ \frac{\left(\varphi_{i+1}-\varphi_i\right)c_{i,i+1}}{W_p} \right]^{2.3} \times \]

\[ \times\left(1-\psi^{4.3}\right)\cdot d_a^2 l \ \mathrm{kg/cm}. \tag{70} \]

It follows from this that the total hysteresis loss \(A_h\) is proportional to the absolute amplitude of oscillation of the first mass \(\varphi_{1abs}\) to the power \(2.3\), i.e., in general form:

\[ A_h=a\cdot \varphi_{1abs}^{\,2.3}, \tag{71} \]

where

\[ a=15.85\cdot 10^{-10}\cdot \sum\left[ \frac{\left(\varphi_{i+1}-\varphi_i\right)\cdot c_{i,i+1}}{W_p} \right]^{2.3} \times \]

\[ \times\left(1-\psi^{4.3}\right)\cdot d_a^2\cdot l. \tag{72} \]

The values of the relative amplitudes \(\varphi_i\) of all the masses of the installation entering into expression (72) are known from the preceding calculation (see above, the calculation according to Widler). But the calculation of the value \(a\) by formula (72) for these \(\varphi_i\), i.e. under the assumption \(\varphi_1=1\) radian, leads to extremely large values of \(\tau_{\max}\), inconvenient for practical computations. Therefore it is more expedient to take here

\[ \varphi_1=\xi \ \text{radian}, \]

where \(\xi\) is a sufficiently small quantity, for example equal to \(1/100\) radian. Then we obtain new values of the relative amplitudes of oscillation of all the masses, with \(\varphi_1=\xi\): for the second mass, \(\xi\varphi_2\); for the third mass, \(\xi\varphi_3\); and in general for the \(i\)-th mass, \(\xi\varphi_i\).

With such amplitudes the hysteresis loss in the shaft, on the basis of formula (69), will be equal to:

\[ A_{h,\xi}=15.85\cdot 10^{-10}\cdot \sum\left[\frac{(\xi\varphi_{i+1}-\xi\varphi_i)\cdot c_{i,i+1}}{W_p}\right]\times \]

\[ \times\left(1-\psi^{4.3}\right)\cdot d_a^2\cdot l \tag{73} \]

The numerical calculation of \(A_{h,\xi}\) by this formula in each individual concrete case presents no difficulties.

On the basis of formula (70), \(A_{h,\xi}\) and \(A_h\) are related by the ratio:

\[ \frac{A_{h,\xi}}{A_h}=\left(\frac{\xi}{\varphi_{1\text{ abs}}}\right)^{2.3}, \tag{74} \]

or

\[ A_h=A_{h,\xi}\cdot \left(\frac{\varphi_{1\text{ abs}}}{\xi}\right)^{2.3}. \tag{75} \]

To determine the greatest value of the forced oscillations \(\varphi_1\) of the first mass of the elastic system under established resonance, it is necessary to equate the work of the exciting moments over an oscillation cycle to the work absorbed by elastic hysteresis:

\[ A_{\text{exc}}=A_h, \]

or, by formulas (41) and (75),

\[ \pi M\cdot \varphi_{1\text{ abs}}\cdot \overrightarrow{\sum}\varphi_i = A_{h,\xi}\cdot \left(\frac{\varphi_{1\text{ abs}}}{\xi}\right)^{2.3}; \]

where, according to formula (7),

\[ M=C_h\cdot F\cdot r. \]

Hence

\[ \varphi_{1\text{ abs}} = \sqrt[1.3]{\frac{\pi M\cdot \overrightarrow{\sum}\varphi\cdot \xi^{2.3}}{A_{h,\xi}}} \ \text{radians}. \tag{76} \]

The actual amplitudes of the resonance oscillations of all the other masses of the installation, in particular the flywheels, just as above, when calculated according to Wydler, are determined by multiplying their relative amplitudes by \(\varphi_{1\text{ abs}}\), calculated by formula (76).

The forced resonance oscillations of all masses lag by \(90^\circ\) or \(270^\circ\) (depending on the sign of the corresponding \(\varphi_{i\text{ abs}}\)) behind the vector \(\overrightarrow{\sum}\varphi_i\), constructed in the phase diagram of the exciting resonating harmonics of the torque (see Fig. 7).

Dorey’s investigations (Dorey)^41. The data of Rausch, used by Lewis in calculations of forced torsional vibrations and forming the basis of formulas (60) and the following ones, were obtained from experiments with only one grade of low-carbon steel and, of course, cannot have universal significance for all those steels that are at present used for the manufacture of crankshafts of internal-combustion engines and, in general, of the main shafts of power plants.

Dorey’s work is a major step forward in the study of hysteresis losses in shafts made of modern grades of steel.

Tables 1 and 2 give, respectively, the general characteristics and the chemical composition of the steels tested by Dorey. (In the last column of Table 1 we give approximate steel grades, equivalent in chemical composition, according to the SAE and OST 7123—7124 classifications.)

Table 1

Name of steel	Steel designation (according to Dorey)	Approximate equivalent (by chemical composition) steel grades according to the SAE and OST 7123—7124 classifications
Carbon . . . .	A	1020—1120 SAE; 20 OST
» . . . .	B	1030 SAE; 30 OST
Nickel (3-percent) .	C	2230 SAE —
Chromium-nickel . . .	D	2435 SAE; 20KhN3OST
Chromium-vanadium . .	E	6145 SAE —

Table 2

Steel designation (according to Dorey)	C	Mn	Si	S	P	Ni	Cr	Va
A	0,21	0,66	0,17	0,026	0,021	0,05	—	—
B	0,30	0,63	0,31	0,031	0,019	0,12	—	—
C	0,28	0,59	0,20	0,007	0,013	3,17	—	—
D	0,31	0,54	0,29	0,014	0,016	3,42	0,85	—
E	0,45	0,71	0,28	0,009	0,017	0,36	1,38	0,17

^41 S. Dorey, Elastic hysteresis in crank—shaft steel, “Proc. of the Inst. of Mech. Engineers, London,” Vol. 123 (1932), p. 479.

Heat treatment of steels:
A and B—ordinary normalizing for steel forgings at 850°;
C—oil quenching, 850°; tempering 610°, cooling in air;
D—oil quenching, 830°; tempering 610°, cooling in water;
E—oil quenching, 850°; tempering 710°, cooling in air.

The mechanical characteristics of the steels under static tensile tests are given in Table 3.

Table 3

Mechanical characteristics	A	B	C	D	E
Proportional limit $\sigma_p$ (in kg/cm²)	2360	2600	1490	3780	4880
Yield point $\sigma_s$ (in kg/cm²)	2570	3230	5300	8160	7940
Ultimate strength $\sigma_B$ (in kg/cm²)	4690	5890	7380	9480	8480
Elongation (in %)	42.0	33.0	27.5	23.5	26.0

As a result of his experiments, Dorey arrives at the following conclusions.

All the steels investigated have a critical point (critical stress $\tau_c$), beyond which the hysteresis loss in the metal increases very rapidly. For the carbon steels A and B, $\tau_c$ is approximately 10–15% below the endurance limit of the metal in fully reversed torsion, and for all practical cases may be taken equal to $0.2\sigma_B$. For the following classes of steels, $\tau_c$ is approximately equal to the endurance limit of the metal in fully reversed torsion.

The energy loss to hysteresis per cycle of oscillation, at any range of stresses, depends on the previous history of the metal. If the metal has previously been subjected to cyclic stresses below $\tau_c$, then the energy loss under subsequent loads with a smaller stress remains the same as for a new metal. Dorey calls this case of loading primary. If, however, the metal has previously been subjected to stresses greater than $\tau_c$, then the energy loss in it per cycle of oscillation, at stresses smaller than $\tau_c$, is greater than for a new metal. This case of loading may be called secondary.

The magnitude of hysteresis loss is almost entirely independent of the vibration speed, which is also confirmed by the experiments of Hopkinson, Rowett, Föppel, Bäcker, and others.

According to Dorey, the hysteresis loss in the shaft per one cycle of oscillation, referred to a unit volume of the shaft, may be determined by the expression (in our notation):

\[ \Delta A_h=a_h\cdot\left(\frac{\tau_{\max}}{\tau_c}\right)^q, \tag{77} \]

where \(\Delta A_h\) is the hysteresis energy (in \(kg\,cm/cm^3\));
\(a_h\) is the hysteresis constant;
\(q\) is the hysteresis exponent;
\(\tau_{\max}\) is the maximum torsional stress in the shaft during oscillation (in \(kg/cm^2\));
\(\tau_c\) is the critical stress (in \(kg/cm^2\)).

The values of \(q\), as well as \(a_h\) and \(\tau_c\), recalculated by us for the metric system, are given in Table 4.

Table 4

Steel designations (according to Dorey)	Primary loading case \(a_h\)	Primary loading case \(q\) at \(\tau<\tau_c\)	Primary loading case \(q\) at \(\tau>\tau_c\)	Secondary loading case \(a_h\)	Secondary loading case \(q\) at \(\tau<\tau_c\)	Secondary loading case \(q\) at \(\tau>\tau_c\)	Critical stress \(\tau_c\) (in \(kg/cm^2\))
A	\(12.7\cdot10^{-3}\)	2.8	10.0	\(52.1\cdot10^{-3}\)	3.2	6.6	945
B	\(12.0\cdot10^{-3}\)	3.0	10.5	\(62.6\cdot10^{-3}\)	3.5	6.7	1100
C	\(51.4\cdot10^{-3}\)	3.2	10.5	\(218.1\cdot10^{-3}\)	3.5	6.7	1860
D	\(25.3\cdot10^{-3}\)	2.7	9.0	\(56.3\cdot10^{-3}\)	2.8	6.5	2330
E	\(7.5\cdot10^{-3}\)	2.8	10.5	\(67.5\cdot10^{-3}\)	3.2	6.4	1730

Table 4 shows that steels with a high elastic limit have small hysteresis losses. But the damping capacity of steel is that essential characteristic of it which must be taken into account when selecting the material for a crankshaft. From this point of view, steel with a large absorption of energy due to elastic hysteresis is especially suitable for

shafts of those engines which, during start-up, must pass through a number of strong critical speeds. In these cases steels A, B, and C proved considerably more advantageous than steels D and E.

Using expression (77) and repeating the derivations of formula (62), one can obtain the following formula for the hysteresis loss in a hollow cylindrical shaft with diameters \(d_a\) and \(d_i\) and length \(l\), for one cycle of vibration:

\[ A_h=\frac{\pi}{2(q+2)}\cdot a_h\cdot \left(\frac{\tau_{\max}}{\tau_c}\right)^q \cdot\left(1-\psi^{q+2}\right)\cdot d_a^2 l, \tag{78} \]

or, as before,

\[ A_h=k_h\cdot d_a^2 l, \]

where

\[ k_h=\frac{\pi}{2(q+2)}\cdot a_h\cdot \left(\frac{\tau_{\max}}{\tau_c}\right)^q \left(1-\psi^{q+2}\right) \]

and

\[ \psi=\frac{d_i}{d_a}. \]

The total hysteresis loss in the shaft of the power plant over a vibration cycle at resonance, when the amplitude of vibration of the first mass is \(\varphi_1=\xi\), will now be determined, similarly to formula (73), by the expression:

\[ A_{h,\xi}=\sum k_h\cdot d_a^2 l= \]

\[ =\sum \frac{\pi}{2(q+2)}\cdot a_h\cdot \left[ \frac{(\xi\varphi_{i+1}-\xi\varphi_i)\cdot c_{i,i+1}} {W_p\cdot \tau_c} \right]^q \cdot \left(1-\psi^{q+2}\right)\cdot d_a^2 l \tag{79} \]

and, analogously to formula (75):

\[ A_h=A_{h\xi}\cdot \left(\frac{\xi}{\varphi_{1\,abs}}\right)^q, \]

where \(A_h\) is the actual hysteresis loss in the shaft at the vibration amplitude \(\varphi_{1\,abs}\) of the first mass.

Finally, from the equality of the work of the exciting moments and the damping forces, using formulas (41) and (79), we obtain the final calculation formula for the actual resonant amplitude of vibration of the first mass of the engine:

\[ \varphi_{1\,abs} = \sqrt[q-1]{\frac{\pi M \sum \cdot \varphi_i \cdot \xi^q}{A_{n,\xi}}} \tag{80} \]

when hysteresis losses are taken into account according to Dorey’s data. This formula is analogous to formula (76).

The actual resonant amplitudes of the forced vibrations of all the remaining masses of the installation are now determined in exactly the same way as in the calculation according to Wydler and Lewis (see above).

When calculating the hysteresis loss in the shaft by formula (79), one should always, under normal conditions, assume the case of primary loading at \(\tau_{\max} < \tau_c\), since the strength conditions of the shaft do not permit allowing excessively large additional stresses in it from torsional vibrations.

General characterization of the methods of Wydler, Lewis, and others. At the present time, not all the data necessary for a completely reliable calculation of the amplitudes of forced vibrations at resonance are yet available, and consequently also for an entirely faultless construction of theoretical torsiograms for the shaft of an engine operating at one of its critical speeds.

Underlying all the methods considered above for determining resonant amplitudes are, first of all, the following assumptions:

1) the curve of the form of the forced vibrations of the system at resonance coincides with the curve of the form of its own free vibrations;

2) the damping forces in the installation do not reduce the frequency of the natural vibrations of its shaft;

3) the greatest amplitudes of the forced vibrations are obtained when the frequency of the resonating harmonic of the torque coincides exactly with the frequency of the natural vibrations of the system;

4) the engine operates for a sufficiently long time at the critical number of revolutions; the number of revolutions is strictly constant, and the resonance is not disrupted.

The first of these assumptions is entirely confirmed (within the limits of practical accuracy) by Wydler’s detailed investigations. The second and third assumptions are also quite acceptable in all practical calculations of resonant torsional vibrations²². The fourth assumption is justified during prolonged operation of the engine in a steady regime, with a good governor and a constant load; moreover, the error introduced by this assump—

…on the average, leads only to somewhat exaggerated, as compared with reality, calculated values of the resonant amplitudes and of the additional stresses in the shaft, i.e. it gives an error on the side of shaft strength.

The greatest difficulties in calculating forced torsional vibrations are presented by the determination of the magnitude of the damping forces and of their character.

The numerical values of the coefficient of velocity resistances \(k\) according to Widler, given above, cannot claim sufficient accuracy. Geiger’s experiments (Geiger)⁴³ showed variations in the value of \(k\) in the internal-combustion engines he investigated within the limits \(1:10\). A similar deviation is also given by Mann’s experiments.

There are grounds for believing that the coefficient \(k\) according to Widler is not a constant quantity for one and the same engine, but varies depending on the regime and the operating conditions of the power plant. Therefore calculations of forced vibrations by Widler’s method, when using values of \(k\) that are constant for a given type of engine, have only relative value and are suitable only for comparative investigations that do not require high accuracy.

The dependence for the coefficient \(k\) obtained by Shannon (see Fig. 11) makes it possible to take the damping forces into account more reliably when determining resonant amplitudes by Widler’s method. The processing of data from numerous experimental investigations of torsional vibrations in installations with internal-combustion engines, carried out by Shannon and presented in his article cited above,⁴⁴ apparently confirms the practically sufficient accuracy, in most cases, of the dimensionless damping coefficient established by him

\[ \frac{\theta \omega_0}{k_0}. \]

Calculation according to Lewis’s theory on the basis of Rowett’s experiments usually gives somewhat exaggerated values of the amplitudes of forced resonant vibrations. Porter⁴⁵ points out cases in which the actual amplitudes of resonant vibrations, found by torsiographing, amounted to \(0.4\) of those calculated according to Lewis—Rowett. This is equivalent to increasing the hysteresis loss by \(3.29\) as against its value determined by formula (60).

Brown, in the discussion of Dorey’s report,⁴⁶ notes investigations of two marine diesel installations, for which the calculation

⁴³ I. Geiger, Dämpfung bei Drehschwingungen von Motoren, “Z. d. V. D. I.”, 1934, p. 1353.

⁴⁴ See footnote 37.
⁴⁵ See footnote 42.
⁴⁶ See footnote 41.

according to Lewis—Rauhut, gave values of the resonance amplitudes exaggerated by 1.95–2.17 times as compared with the actual ones. N. Urvantsev also points to the possibility of analogous discrepancies,^47 noting that a large difference between the actual and calculated (according to Lewis—Rauhut) amplitudes occurs in ship installations with a propeller in the case of two-node vibrations of the main shaft.

One of the causes of such a discrepancy between the calculation data according to Lewis—Rauhut and the results of direct torsiographing of an operating installation may be considered to be the inaccuracy of formula (60) for the materials of modern crankshafts. The use of Dorey’s formula for hysteresis loss in the shaft (formula 77) must undoubtedly considerably increase the reliability of the calculation of resonance amplitudes by Lewis’s method.^48 But even in this case a rather large discrepancy between theoretical calculation and experiment is sometimes possible.^49

Lewis’s theory is based on taking into account only the hysteresis loss in the shaft. In fact, however, along with elastic hysteresis there is a whole series of other sources of damping forces in the power installation.

In accordance with Geiger,^50 all damping forces arising during torsional vibrations of an engine shaft may be divided into real (Wirksame Dämpfung) and apparent (Scheinbare Dämpfung). Real damping forces reduce the amplitudes of forced resonance vibrations by converting the work of the exciting harmonic moments into heat. Apparent damping causes a reduction of the resonance amplitudes without loss of energy (it occurs in every piston engine).

The reduced mass of the connecting-rod-and-crank mechanism of each cylinder of the engine varies with the angle of rotation of the crankshaft. It reaches a minimum at the top and bottom dead centers, and a maximum at crank angles of 90° and 270°.^51 In accordance with this change in the reduced masses in the engine, the frequency of the natural vibrations of the shafting also changes. Therefore, theoretically, complete resonance in the shaft of a power installation with a piston engine can never be attained. The constant disturbance of resonance leads to a decrease in the resonance ampli-

^47 N. Urvantsev, Critical Speeds in Diesel Installations, 1931.

^48 The work by E. Lunts, deserving of great attention, “On the Determination of Damping Forces from Internal Friction in Torsional Vibrations of Metallic Rods” (Vestnik metallopromyshlennosti, 1936, Nos. 5, 12–13, 15), requires the development of a special methodology for calculating resonance amplitudes, consideration of which lies beyond the scope of the present article.

^49 W. Ker. Wilson. Engineering, 1934, p. 167.

^50 “Z. d. V. D. I.”, 1934, p. 1353.

^51 See, for example, I. Geiger, Mechanische Schwingungen, Berlin, 1927.

here compared with those calculated theoretically under the usual assumption of an unchanging value of the reduced masses of the engine.

The actual damping in the engine is due to:

1) additional friction forces on the journals of the crankshaft, arising during its torsional vibrations;

2) additional friction in the guides;

3) loss of energy through impacts in bearings, flanges, couplings, keyed joints, etc.;

4) transmission of energy into the surrounding medium and the foundation, arising during vibrations of the machine;

5) loss of energy due to the elastic hysteresis of the shaft.

According to Geiger’s experiments, the decisive factor is the hysteresis loss in the shaft. It amounts to 62–64% of all losses at strong critical speeds and 16–35% at weaker (secondary) critical speeds.

The hysteresis loss can be considerably increased (in order to reduce resonant amplitudes and stresses in the shaft) by introducing an intermediate shaft between the engine and the generator, provided that this shaft has a somewhat smaller diameter than the crankshaft journals. To obtain the necessary effect, even a small difference in the diameters of the two shafts is sufficient, since the hysteresis loss according to Dorey, for $\tau_{\max} < \tau_c$, is approximately proportional to $\tau_{\max}^{3}$, while $\tau_{\max}$ is inversely proportional to $d^{3}$. Consequently, the hysteresis loss per unit volume of the shaft is inversely proportional approximately to the ninth power of $d$.

The additional work of friction in the pins, guides, etc., caused by the additional loads in them, amounts, according to Geiger, to 30% of the total loss at weak (secondary) critical speeds and 3–6% at strong (principal) critical speeds.

Apparent damping is equivalent to a loss of 20% and 6–7%, respectively, at weak and strong critical speeds.

The other sources of losses in the engine indicated above play an extremely insignificant role.

In marine installations, the damping forces acting on the propeller may be a very substantial source of losses. This loss may be neglected only if the node of vibration is located close to the propeller and the vibration amplitudes of the latter are small.

To take account of the work of all the damping forces in the installation, Porter, in the discussion of Hartog’s paper (L. P. den Hartog),²³ proposed the Lewis—Rautt formula (60), determining the hystere-

...frictional loss in a thin-walled shaft, referred to a unit of its volume, by another:

\[ \Delta A_h = 4.5 \cdot 10^{-10} \cdot \tau_{\max}^{2.3}\ \text{ft. dm}/\text{dm}^3, \tag{81} \]

Porter’s formula, in metric measures, takes the form:

\[ \Delta A_h = 142 \cdot 10^{-10} \cdot \tau_{\max}^{2.3}\ \text{kg cm}/\text{cm}^3, \tag{82} \]

where \(\tau_{\max}\) is in \(\text{kg}/\text{cm}^2\).

The increase of the numerical coefficient in formula (82), as compared with formula (61), must take into account all losses in the installation (with the exception of losses at the propeller, during its large oscillations). The value of the correction coefficient, \(\frac{4.5}{1.37}\), was found by Porter experimentally. A comparison of the results of calculation by formula (81) with experimental data, carried out by Porter for a whole series of power installations, confirms its sufficient reliability.

The use of formula (82), instead of formula (61), does not change the calculation method according to Lewis considered above. It is sufficient only, in expression (76), to increase \(A_{h,\xi}\) by \(\frac{4.5}{1.37} = 2.84\) times. Of course, Porter’s formula is valid only in the absence of clearances in shaft couplings, clutches, etc.

An analysis of all the data set forth above on damping forces arising during torsional vibrations of shafts emphasizes the unconditional necessity of further experimental work on the study and differentiation of the sources of energy loss in an oscillating system.

The calculation of forced resonant vibrations according to Widler or Lewis is not capable of giving completely reliable results, since the variety of damping forces in a power installation cannot be expressed by a single formula. The laws of variation of friction forces in the engine, of damping forces at the propeller, etc., are entirely different from the laws of elastic hysteresis.

Only a separate analysis of all, or at least the most important, damping forces will provide the necessary greater accuracy of calculations of forced torsional vibrations in the shafts of power installations.

V. THEORETICAL TORSIOGRAMS AND THE ACTUAL DEGREE OF NONUNIFORMITY OF ROTATION

The method, set forth in the preceding sections, for determining the amplitudes of forced torsional vibrations of a shaft line provides all the data necessary for constructing theoretical torsiograms for the flywheel or the rotor of a dynamo.

Theoretical torsiograms, plotted, for example, for the rotor of a dynamo, make it possible:

1) determine the angles of lag or advance of the rotor relative to uniform rotation;

2) construct the theoretical tachogram, i.e. the curve of the actual rotational velocities of the rotor;

3) determine the actual degree of nonuniformity of rotation of the rotor;

4) determine the actual magnitude of the maximum torsional stresses in the shaft between the engine and the generator.

The construction of theoretical torsiograms for a number of different shaft speeds makes it possible to establish forbidden zones and the most advantageous zones of engine operating speeds from the dynamic point of view.

Comparison of theoretical torsiograms with actual ones, obtained by direct torsiographing of the given operating installation, serves as a means of checking the degree of accuracy of the theoretical calculations of forced torsional vibrations.

Finally, comparison and joint analysis of the theoretical and the actual torsiograms makes it possible to study in detail the behavior of the damping forces in the machine, to determine their specific magnitude, and to investigate their dependence on the operating regime of the engine, etc.

Each theoretical torsiogram is constructed for a definite number of engine revolutions and for a definite mass of the installation. Let \(n\) be the number of engine revolutions at which the torsiogram is to be constructed for the flywheel or for the rotor of the dynamo. In this case, at \(n\), as at any other, the true law of motion of each mass of the installation is characterized by the superposition, upon the uniform rotation of the shaft with mean angular velocity, of all those forced harmonic vibrations that are caused by all harmonic components of the torque.

But for the practical construction of a theoretical torsiogram there is no need to take into account very many weak harmonic vibrations. One should take into account the influence of the harmonic moments of the principal, especially the lower, orders, resonating both above and below the given shaft speed \(n\), and then the influence of secondary harmonics that give critical speeds close to the operating \(n\).

Let \(h\) be the orders of the torque harmonics to be taken into account. Each of these harmonics causes forced torsional vibrations of the entire shafting with angular velocity:

\[ \omega = h \eta\ \mathrm{sec}^{-1} \]

and with frequency:

\[ N = hn . \]

Here \(\eta=\dfrac{\pi n}{30}\) is the angular velocity of rotation of the shaft for the specified \(n\).

Next, for each of the selected harmonics the following calculations are carried out.

We compute the parameter:

\[ A=\frac{\theta \omega^{2}}{c}=\frac{\theta(h\eta)^{2}}{c}, \]

and from it, using the appended tables or formulas (11), valid for an in-line engine with uniform distribution of its masses along the crankshaft, we find the relative amplitudes of oscillation of the engine masses. The relative amplitudes of oscillation of the flywheel and of the dynamo rotor are computed from formulas (12) and (13) when \(h\eta \ne \omega_{0}\), and from formulas (49), (50) when \(h\eta=\omega_{0}\).

If the harmonic under investigation is not resonant at the given \(n\), then from formulas (14), (15) we compute \(R\), construct the phase diagram, determine from it \(\vec{\Sigma\varphi_i}\) and the angle \(\gamma\), and from formula (19) find \(\varphi_{\mathrm{I}\,abc}\) or \(\varphi_{\mathrm{II}\,abc}\). The harmonic oscillatory motion of the flywheel is then represented by equation (25).

The amplitudes of the oscillations produced by resonant \((h\eta=\omega_{0})\) harmonic moments are computed by the methods of Widder or Lewis. In calculation by Widder’s method it is advisable to take the damping forces into account according to Shannon’s data. In calculation by Lewis’ method one should use Dorey’s data or Porter’s empirical formula. The phase angle \(\gamma_{0}\) of the resonant oscillations is found from the phase diagram, according to the directions given in Section III of the present article.

In a similar way, the amplitudes, phase angles, and angular velocities \((h\eta)\) of the separate component harmonic oscillations of the flywheel or of the dynamo rotor are determined. The period of each of these oscillations is equal to the duration of one revolution of the shaft divided by the order of the harmonic moments causing these oscillations.

The construction of the sinusoids of all the computed component oscillations and the summation of their corresponding ordinates gives the desired theoretical torsiogram for the given rotational speed of the engine shaft. The ordinates of the torsiogram represent the magnitudes of the angles of advance or lag of the mass relative to its uniform rotation with mean angular velocity \(\eta\), at each instant of time. Figure 12 shows the construction of a torsiogram for the flywheel of a six-cylinder engine.

Fig. 12

The instantaneous additional angular velocity, superposed on the uniform rotation and caused by torsional vibrations of the shaft, is characterized, obviously, by the tangent of the angle of inclination of the tangent to the torsiogram drawn at the given point of it²⁴. But the construction of these tangents is difficult and cannot ensure the necessary accuracy of the results. For analyzing the regularities in the variation of the actual rotational speed of the flywheel or of the dynamo rotor, it is advisable to construct a new curve—the theoretical tachogram.

In the general case, the motion of the flywheel represented by the torsiogram is analytically expressed by:

\[ \alpha_{m}=\eta t+\sum \varphi_{\mathrm{I}\,\mathrm{abs}}\cdot \sin(h\eta t+\gamma), \tag{83} \]

where \(\alpha_{m}\) are the absolute angles of rotation of the flywheel at each instant of time relative, for example, to a plane passing through the cylinder axes;

\(\eta\) is the mean angular velocity of rotation of the shaft;

\(\varphi_{I\,abs}\) are the amplitudes of the forced oscillatory motions of the flywheel.

Differentiation of formula (83) with respect to time gives the equation for the actual absolute angular velocity of rotation of the flywheel:

\[ \eta_m=\frac{d\alpha_m}{dt}=\eta+\sum \varphi_{I\,abs}\cdot h\eta\cdot \cos(h\eta t+\gamma)= \]

\[ =\eta+\sum \varphi_{I\,abs}\cdot h\eta\cdot \sin(h\eta t+\gamma'), \]

or

\[ \eta_m=\eta+\sum \eta_h\cdot \sin(h\eta t+\gamma'), \tag{84} \]

where

\[ \eta_h=\varphi_{I\,abs}\cdot h\eta \tag{85} \]

is the maximum velocity of the forced oscillatory motion of the flywheel, produced by the harmonic components of the \(h\)-th order of the torque on the engine shaft. This velocity \(\eta_h\) should not be confused with the angular velocity \(h\eta\) of the forced harmonic oscillations of the flywheel.

In formula (84)

\[ \gamma'=\gamma+90^\circ, \tag{86} \]

i.e., the vectors \(\eta_h\) of the harmonic oscillatory motions of the flywheel lead by \(90^\circ\) the vectors of the angles of deviation \(\varphi_{I\,abs}\), but, of course, have the same angular velocity of their rotation and the same period.

The construction of all sinusoidal curves (taking account of their phase angles \(\gamma'\)), determined by the expressions²⁵:

\[ (\eta_h)=\eta_h\cdot \sin(h\eta t+\gamma') \]

and entering under the summation sign in formula (84), and the summation of their ordinates gives the required theoretical tachogram for the flywheel (Fig. 13). In a completely similar way, a theoretical tachogram may also be constructed for the dynamo rotor, in the arrangement according to Fig. 3.

Plotting downward from the mean line \(Ot\) (the abscissa axis) of the tachogram the mean angular velocity \(\eta\) of rotation of the shaft, we obtain a new axis \(O_1t\)—the line of zero absolute velocities of rotation of the flywheel. The ordinates of the tachogram, measured to this new axis, give the absolute magnitude of the angular velocity of rotation of the flywheel—

Fig. 13

Labels in the figure: \(5\eta\), \(6\eta\); “1 revolution of the shaft”; \(0\), \(\tau\), \(\eta\), \(\eta_{\min}\), \(\eta_{\max}\).

of the flywheel at each moment of time. Now the actual degree of nonuniformity of rotation of the flywheel is determined by formula (1):

\[ \delta = -\,\frac{\eta_{\max}-\eta_{\min}}{\eta}, \]

where \(\eta_{\max}\) and \(\eta_{\min}\) are, respectively, the maximum and minimum ordinates of the tachogram (see Fig. 13).

Obviously, this value of $\delta$ for the flywheel, as well as the theoretical torsiogram and tachogram constructed earlier for it, are valid only for one specified value, adopted in the calculation, of the forced torsional vibrations of the shaft speed $n$. Even a comparatively small change in $n$ may have a very strong effect on the character of the torsiogram, the tachogram, and the value of $\delta$.

Let us first assume that the torque on the engine shaft, produced in one cylinder, is represented by the expression:

\[ M=M_0+M_h\cdot \sin (h\eta t+\beta_h), \tag{87} \]

i.e., it consists of a mean constant moment $M_0$ and one harmonic moment $M_h$ of order $h$. Under these conditions the flywheel moves according to the law:

\[ \alpha_m=\eta t+\varphi_{\mathrm{I}\,\mathrm{abs}}\cdot \sin'(h\eta t+\gamma), \tag{88} \]

with angular velocity:

\[ \eta_m=\frac{d\alpha_m}{dt} =\eta+\varphi_{\mathrm{I}\,\mathrm{abs}}\cdot h\eta\cdot \sin (h\eta t+\gamma'), \tag{89} \]

where $\gamma'$ is determined by formula (86). Now the theoretical tachogram for the flywheel will be a simple sinusoid (Fig. 14). The maximum velocity of the harmonic oscillatory motion of the flywheel is equal to the amplitude of this sinusoid:

Fig. 14

\[ \eta_h=\varphi_{\mathrm{I}\,\mathrm{abs}}\cdot h\eta; \]

and, correspondingly, the maximum and minimum absolute angular velocities of its rotation will be:

\[ \eta_{\mu\max}=\eta+h\eta\varphi_{\mathrm{I}\,abs}, \]

\[ \eta_{\mu\min}=\eta-h\eta\varphi_{\mathrm{I}\,abs}. \]

Hence the degree of nonuniformity of rotation of the flywheel is:

\[ \delta_h=\frac{\eta_{\mu\max}-\eta_{\mu\min}}{\eta}=2h\varphi_{\mathrm{I}\,abs}. \tag{90} \]

Thus, the degree of nonuniformity \(\delta_h\) of rotation of the flywheel, caused by the action on the shaft of harmonic moments of only one order, is directly proportional to the amplitude of oscillation of the flywheel \(\varphi_{\mathrm{I}\,abs}\) and to the order of the harmonic causing these oscillations.

The quantity \(\varphi_{\mathrm{I}\,abs}\) in formulas (88), (89), (90) is a function of the angular velocity of the forced oscillations (or, what is the same, of the angular velocity of the exciting harmonic moment), i.e., it depends on the rotational speed of the shaft \(\eta\), or on its number of revolutions.

Using formula (19), one can, for various \(n\), compute \(\varphi_{\mathrm{I}\,abs}\) produced by harmonic moments of one order (Fig. 15). The maximum ordinate of the curve \(\varphi_{\mathrm{I}\,abs}=f(n)\)

Fig. 15

in Fig. 15, which is the resonance value of \(\varphi_{\mathrm{I}\,abs}\) at the critical number of revolutions of the engine:

\[ n_{cr}=\frac{N_0}{h}, \tag{91} \]

where \(N_0\) is the frequency of the natural oscillations of the shafting and must be determined separately, by the method of Widler, Lewis, etc. In the absence of damping forces in the installation, the curve \(\varphi_{\mathrm{I}\,abs}=f(n)\) has a discontinuity at the point corresponding to \(n_{cr}\).

Multiplying all the ordinates of the curve in Fig. 15 by \(2h\), in accordance with formula (90), gives the values of \(\delta_h\) of the flywheel, produced

...by harmonics \(M_h\) of a given order for various \(n\) (Fig. 17).

Under real conditions the torque on the shaft, according to formula (6), is composed of the mean torque \(M_0\) and a series of harmonic torques of various orders. On the basis of concrete calculations of the dependences \(\varphi_{\text{abs}}=f(n)\) and \(\delta_h=f_1(n)\), a diagram may be constructed for the actual amplitudes of the component harmonic oscillations of the flywheel (Fig. 16), and a diagram for the degrees of nonuniformity of its rotation corresponding to these oscillations (see Fig. 17).

Fig. 16

The total degree of nonuniformity \(\delta\) of the flywheel rotation is determined only from the construction of the theoretical tachogram for each specified number of revolutions. But in any case the actual \(\delta\) will be no greater than the sum of the ordinates of all the curves in Fig. 17 measured for the corresponding \(n\).

Fig. 17 indicates a considerably more complex dependence between the flywheel \(\delta\) and the engine \(n\) than is established by the classical formula (2).

Fig. 17

V. Critical and Optimal Values of \(GD^{2}\) of the Flywheel

Formula (2) asserts an inversely proportional dependence between the degree of nonuniformity of rotation \(\delta\) and the flywheel moment \(GD^{2}\) of the flywheel. From this formula one usually draws, in a purely formal way quite natural and based on the theory of calculating \(\delta\) from the diagram of tangential forces or by Wittenbauer’s method, the conclusion that there is direct proportionality between \(GD^{2}\) and the effectiveness of the flywheel from the standpoint of improving the uniformity of rotation of the machine shaft. But, as indicated above, practice in a number of cases refutes this proposition and requires the introduction of the concept of harmful flywheel masses and of the critical value of \(GD^{2}\) of the flywheel.^55

The method set forth above for analyzing forced torsional oscillations makes it possible to establish the actual relation between the \(GD^{2}\) of the flywheel and \(\delta\) in each particular case, for each specified number of revolutions of the engine crankshaft.

The harmonic moments \(M\) of one order \(h\), acting on the engine masses, cause forced oscillations of the flywheel with amplitude \(\varphi_{\mathrm{I}\,abs}\), determined by formula (19), which is valid for oscillations outside resonance. The residual moment \(R\), entering into formula (19), may, on the basis of formulas (12) and (14), be represented in the form:

\[ R=-Ac\cdot\left[\alpha_{\mathrm{I}}\varphi_m+ \left(1-\frac{\alpha_{\mathrm{I}}}{\beta_1}A\right) \sum_{1}^{m}\varphi_i\right] \tag{92} \]

for the arrangement according to Fig. 2. Substitution of this value into formula (19) gives:

\[ \varphi_{\mathrm{I}\,abs} = - \frac{ M\sum \varphi_i }{ Ac\cdot \left[ \left(1-\frac{\alpha_{\mathrm{I}}}{\beta_1}A\right) \sum_{1}^{m}\varphi_i+\alpha_{\mathrm{I}}\varphi_m \right] } \]
\[ = - \frac{ M\sum \varphi_i }{ Ac\cdot \left[ \sum_{1}^{m}\varphi_i - \alpha_{\mathrm{I}} \left( \frac{A}{\beta_1}\sum \varphi_i-\varphi_m \right) \right] } , \]

^55 See the work by F. Vogt cited in footnote 8, as well as the work by E. Lvov, “Investigation of the Influence of the Distribution of the Engine Flywheel Masses on the Uniformity of Its Running,” published in Bulletin of the Metal Industry, 1927, No. 11.

where \(\alpha_1\) and \(\beta_1\) are constant for the given particular installation, while the quantities

\[ \vec{\sum}\varphi_i,\quad \sum_{1}^{m}\varphi_i,\quad \varphi_m,\quad A=\frac{\theta\omega^2}{c} \]

depend on the angular velocity of the forced vibrations:

\[ \omega=h\eta=h\cdot\frac{\pi n}{30}, \]

i.e., on the order \(h\) of the exciting harmonics \(M\) and on the rotational speed of the shaft \(\eta\).

For a constant number of engine revolutions \(n\) and a given \(h\), the quantities

\[ \vec{\sum}\varphi_i=\mathrm{const},\quad \sum_{1}^{m}\varphi_i=\mathrm{const},\quad \varphi_m=\mathrm{const},\quad A=\mathrm{const}, \]

\(M=\mathrm{const}\), \(\omega=h\eta=\mathrm{const}\), have quite definite numerical values, independent of the magnitude of the moment of inertia \(\theta_{\mathrm{I}}\) of the flywheel and of the stiffness \(c_{m\mathrm{I}}\) of the shaft connecting the flywheel with the engine.

Under these conditions \((n=\mathrm{const},\ h=\mathrm{const})\), formula (92) may be written as follows:

\[ \varphi_{\mathrm{I\,abs}}=\frac{B_1}{1-\alpha_1B_2}, \tag{93} \]

where \(B_1\) and \(B_2\) are constant quantities determined by the formulas:

\[ B_1=\frac{M\vec{\sum}\varphi_i}{Ac\sum_{1}^{m}\varphi_i}, \tag{94} \]

\[ B_2=\frac{A}{\beta_1}-\frac{\varphi_m}{\sum_{1}^{m}\varphi_i}. \tag{95} \]

Now, from formula (90),

\[ \delta_h=2h\varphi_{\mathrm{I\,abs}}=2h\frac{B_1}{1-\alpha_1B_2}. \tag{96} \]

Expressions (93) and (96) show that the amplitudes of the forced oscillations of the flywheel and the degree of nonuniformity of its rotation depend to a very great extent on the magnitude \(\alpha_I\), i.e., for a given engine, on the magnitude \(\theta_I\) of the flywheel.

In the complete absence of the flywheel \((\theta_I=0)\)

\[ \alpha_I=0, \]

and

\[ \varphi_{I\,abs}=B_1= \frac{M\sum \varphi_i}{Ac\sum \varphi_i} \tag{97} \]

\[ \delta_h=2hB_1. \tag{98} \]

Calculations of \(\varphi_{I\,abs}\) and \(\delta_h\) by formulas (93) and (96) for various values of \(\alpha_I\) make it possible to construct the diagram in Fig. 18.

Fig. 18

For

\[ \alpha_{cr}=\frac{1}{B_2}, \tag{99} \]

i.e., for the value \(\theta_I\) of the flywheel:

\[ \theta_{I\,cr}=\alpha_{I\,cr}\cdot\theta=\frac{\theta}{B_2}, \tag{100} \]

formula (93) gives

\[ \varphi_{I\,abs}=\infty, \]

which is an obvious characteristic of resonance without resistances. The actual value, under practical conditions, \(\varphi_{1\,abs.}\), for

\[ \theta_1=\frac{\theta}{B_2}, \]

must be calculated by formula (51), determining the value entering into it—the amplitude of the forced resonant oscillations of the first mass of the engine, \(\varphi_{1\,abs.}\), according to the instructions of Section III of the present work.

Relations (99) and (100), corresponding to the highest point of the curve \(\varphi_{1\,abs.}\) and \(\delta_h\) in Fig. 18 and making the denominators of expressions (93) and (96) vanish, give the critical value of the flywheel moment of inertia:

\[ \theta_{I\,cr}=\frac{\theta}{B_2} =\frac{\theta}{\dfrac{A}{\beta_1}-\dfrac{\varphi_m}{\displaystyle\sum_1^m \varphi_i}} \tag{101} \]

and the critical flywheel moment:

\[ GD^2_{cr}=4g\cdot\theta_{I\,cr} =4g\,\frac{\theta}{\dfrac{A}{\beta_1}-\dfrac{\varphi_m}{\displaystyle\sum_1^m \varphi_i}}, \tag{102} \]

where \(g\) is the acceleration of gravity;
\(\theta\) is the moment of inertia of the reduced mass of the mechanism of one engine cylinder.

A flywheel with the critical value of the moment of inertia \(\theta_{I\,cr}\) makes the specified number of revolutions of the engine one of the critical rotational speeds of its shaft and gives the worst degree of rotational nonuniformity \(\delta_h\). Fig. 18 and formula (98) show that an engine with no flywheel at all possesses considerably greater uniformity of shaft rotation than with a flywheel having a critical flywheel moment, or one close to it, \(GD^2\).

Obviously, this last conclusion is valid only from the standpoint of the magnitude of the periodic oscillations of the rotational speed of the crankshaft during each of its revolutions, under constant load on the engine. Complete elimination or a considerable reduction of the flywheel \(GD^2\) worsens and often makes altogether impossible the operation of the engine under sudden changes of load. If the flywheel moment is insufficient, the engine stalls in the event of a sudden increase in resistance—for example, in automobiles and tractors at the moment of engaging—

the starting of the friction clutch after starting. From these considerations alone, every piston engine must have a sufficiently large flywheel.^56

The horizontal line drawn in Fig. 18 through point \(a\), corresponding to \(\alpha_{\mathrm{I}}=0\), intersects the right branch of the curve \(\delta_h=f(\alpha_{\mathrm{I}})\) at point \(b\). The hatched segment \(ab\) of this curve indicates the region of harmful values of \(\alpha_{\mathrm{I}}\), i.e., harmful values of the moment of inertia of the flywheel

\[ \theta_{\mathrm{I}}=\alpha_{\mathrm{I}}\theta. \]

Only the choice of a flywheel with moment of inertia

\[ \theta_{\mathrm{I}}>\alpha'_{\mathrm{I}}\theta, \tag{103} \]

or with flywheel moment

\[ GD^2>\alpha'_{\mathrm{I}}\cdot\theta\cdot 4g, \tag{104} \]

where \(\alpha'_{\mathrm{I}}\) is the abscissa of point \(b\) in Fig. 18, can ensure a reduction of \(\delta_h\). Installing on the engine shaft a flywheel having the flywheel moment

\[ GD^2<\alpha'_{\mathrm{I}}\cdot 4g\theta, \]

is not only pointless, but also harmful under the condition that harmonic moments \(M\) of only one order \(h\) act on the shaft, since it leads to an increase in \(\delta_h\), i.e., in the degree of nonuniformity of rotation of the end of the shaft under the flywheel.

As a rule, harmonic moments of different orders act on the engine shaft. Therefore formula (103) and Fig. 18

Fig. 19

^56 See, for example, E. Lvov, Tractors, vol. I, 1933; E. Lvov, Dynamics of Piston Engines, 1936.

usually cannot be used for a concrete determination of the optimal value of \(\Theta_I\) or \(GD^2\) of the flywheel that would ensure the smallest possible value of the full degree of nonuniformity of rotation \(\delta\) in the given power installation. Here it is necessary, using formulas (93) and (96), to calculate the relationships between \(\varphi_{I\,\mathrm{abs}}\) and \(\alpha_I\) for the most important orders of the harmonics of the torque (at a specified rotational speed of the engine shaft) and to construct the diagrams according to Figs. 19 and 20. These diagrams, which are the superposition of the curves in Fig. 18 calculated for different \(h\), show that each order of the torque harmonics corresponds to separate, special, concrete values of \(\alpha_{I\,\mathrm{cr}}\) and \(\Theta_{I\,\mathrm{cr}}\).

Fig. 20

The summation of the amplitudes of the component harmonic oscillations presented in the diagram \(\varphi_{I\,\mathrm{abs}}\) (Fig. 19) can be performed only by constructing theoretical torsiograms for each individual value of \(\alpha_I\). On the basis of the torsiograms, the curve \(\varphi_{I\,\mathrm{abs}} = f(\alpha_I)\) is constructed for the full actual maximum angles of lead or lag of the flywheel, or of the rotor of a flywheel-type generator, relative to its uniform rotation at the mean angular velocity.

In the same way, the construction of theoretical tachograms for various \(\alpha_I\) gives the full value of the true degree of nonuniformity of rotation of the flywheel \(\delta\) and leads to the diagram according to Fig. 21.

The diagrams obtained provide every possibility for the accurate and, undoubtedly, reliable selection of the most advantageous value of the flywheel moment of the flywheel, satisfying all the requirements that, in each concrete case, are imposed by practice with respect to the degree of nonuniformity of rotation of the shaft of the power installation and the angles of lag and lead of the flywheel or of the dynamo rotor.

To save time and reduce the computational work, it is possible, sometimes without great error, to omit the above-mentioned

the construction of theoretical torsiograms and tachograms for various values of \(\alpha_{\mathrm{I}}\), and to calculate the ordinates of the curves \(\varphi_{\mathrm{I}\,\mathrm{abs}}\) and \(\delta\) (see Fig. 21) as arithmetic sums of the ordinates of the component curves \(\varphi_{\mathrm{I}\,\mathrm{abs}}\) and \(\delta\), taken at each value of \(\alpha_{\mathrm{I}}\), respectively, from Figs. 19 and 20. The total degree of nonuniformity of flywheel rotation \(\delta\), for the given \(\theta_{\mathrm{I}}\), will, of course, be no greater than the sum of the ordinates of the curves \(\delta_h\) in Fig. 20, taken for

\[ \alpha_{\mathrm{I}}=\frac{\theta_{\mathrm{I}}}{\theta}. \]

Fig. 21

The method considered for determining the critical and optimal value of the flywheel moment of inertia in the installation according to Fig. 2 is also applicable to the engine—flywheel—dynamo installation according to Fig. 3. In this case formula (19) for the absolute amplitude of the forced harmonic oscillations of the extreme right-hand mass of the reduced elastic system becomes

\[ \varphi_{\mathrm{II}\,\mathrm{abs}}=\frac{M}{R}\sum \varphi_i, \]

where, according to formula (15),

\[ R=-Ac\left(\sum_{1}^{m}\varphi_i+\alpha_{\mathrm{I}}\varphi_{\mathrm{I}}+\alpha_{\mathrm{II}}\varphi_{\mathrm{II}}\right), \]

or, substituting here the values for \(\varphi_{\mathrm{I}}\) and \(\varphi_{\mathrm{II}}\), respectively from (12) and (13),

\[ R=-Ac\left\{ \sum_{1}^{m}\varphi_i+ \alpha_{\mathrm{I}}\left(\varphi_m-\frac{A}{\beta_1}\sum_{1}^{m}\varphi_i\right)+ \right. \]

\[ \left. +\alpha_{\mathrm{II}}\left[ \varphi_m-\frac{A(\beta_1+\beta_2)}{\beta_1\beta_2}\sum_{1}^{m}\varphi_i \right]+ \right. \]

\[ +\alpha_{\mathrm I}\alpha_{\mathrm{II}}\left[ \frac{A^{2}}{\beta_{1}\beta_{2}}\sum_{1}^{m}\varphi_i -\frac{A}{\beta_{2}}\varphi_m \right]\right\}. \]

Now the formula for \(\varphi_{\mathrm{II}\,abs}\) can be written in abbreviated form as:

\[ \varphi_{\mathrm{II}\,abs}=\frac{B_1}{B_3-\alpha_{\mathrm I}B_4}, \tag{105} \]

where

\[ B_1=-\frac{M\overrightarrow{\sum}\varphi_i}{Ac}, \tag{106} \]

\[ B_3=\sum_{1}^{m}\varphi_i+\alpha_{\mathrm{II}}\left[ \varphi_m-\frac{A(\beta_1+\beta_2)}{\beta_1\beta_2}\sum_{1}^{m}\varphi_i \right], \tag{107} \]

\[ B_4=-\left[ \varphi_m+\frac{A}{\beta_1}\left(\frac{\alpha_{\mathrm{II}}A}{\beta_2}-1\right)\sum_{1}^{m}\varphi_i -\alpha_{\mathrm{II}}\varphi_m\frac{A}{\beta_2} \right], \tag{108} \]

or else

\[ \varphi_{\mathrm I\,abs}=\frac{B_1}{B_5-\alpha_{\mathrm{II}}B_6}. \tag{109} \]

Here

\[ B_5=\sum_{1}^{m}\varphi_i+\alpha_{\mathrm I}\left( \varphi_m-\frac{A}{\beta_1}\sum_{1}^{m}\varphi_i \right), \tag{110} \]

\[ B_6=-\left[ \varphi_m-A\frac{(\beta_1+\beta_2)}{\beta_1\beta_2}\sum_{1}^{m}\varphi_i+ \right. \]

\[ \left. +\alpha_{\mathrm I}\left( -\frac{A^2}{\beta_1\beta_2}\sum_{1}^{m}\varphi_i +\varphi_m\frac{A}{\beta_2} \right) \right], \tag{111} \]

and \(B_1\) is determined by formula (106).

Hence, analogously to formula (96), the degree of nonuniformity of rotation of the dynamo rotor, caused by the action on the engine shaft of harmonic torques of the \(h\)-th order:

\[ \delta_h = 2_h \frac{B_1}{B_3 - \alpha_{\mathrm I} B_4}, \tag{112} \]

or

\[ \delta_h = 2h \frac{B_1}{B_5 - \alpha_{\mathrm{II}} B_6}. \tag{113} \]

The coefficients \(B_1, B_2 \ldots B_6\) in formulas (105), (106), (112), (113) have a quite definite constant numerical value for the particular installation under investigation, for the given order \(h\) of the harmonic \(M\), and for the specified constant number of revolutions \(n\) of the engine shaft. The coefficients \(B_1, B_3\), and \(B_4\) are entirely independent of the magnitude of the moment of inertia \(\theta_{\mathrm I}\) of the flywheel. Likewise, \(B_5, B_6\) are independent of the moment of inertia \(\theta_{\mathrm{II}}\) of the generator rotor.

Formula (112) is analogous to formula (96); therefore, using it, one can construct diagrams for \(\delta_h\) as a function of \(\alpha_{\mathrm I}^{*} = \dfrac{\theta_{\mathrm I}}{\theta}\), similar to Figs. 19, 20, 21, and, in accordance with what was said above, find the critical and optimum values of \(GD^2\) of the flywheel in the installation of Fig. 3.

In exactly the same way, on the basis of formula (113), one investigates the dependence between the flywheel moment, or the moment of inertia \(\theta_{\mathrm{II}} = \alpha_{\mathrm{II}}\theta\), of the generator rotor and the degree of nonuniformity of its rotation, and determines the optimum value of its \(GD^2\).

The method set forth here for determining the necessary and most advantageous value of \(GD^2\) of the flywheel or dynamo rotor, for a specified degree of nonuniformity of its rotation or a limited angle of its lag or lead, will require a rather large amount of computational work, but it ensures considerably greater accuracy in solving the problem than is possible when applying the classical methods of calculation by the tangential-force diagram or by the Wittenbauer method.

VI. CLASSICAL DEGREE OF NONUNIFORMITY OF ROTATION \(\delta_s\)

From the method of calculating forced vibrations and constructing theoretical torsiograms set forth above, as a special consequence there follows a method for determining the classical degree of nonuniformity of rotation \(\delta_s\).

The classical methods of flywheel calculation assume absolute torsional rigidity of the shaft of the installation. Under this assumption the amplitudes of vibration of all masses of the shafting (relative to its uniform rotation with the mean angular velocity) are always identical to one another:

\[ \varphi_1=\varphi_2=\ldots=\varphi_m=\varphi_{\mathrm I}=\varphi_{\mathrm{II}}, \]

since it is assumed that \(c=\infty,\ c_{m\mathrm I}=\infty,\ c_{\mathrm{I,II}}=\infty\).

Now from formula (15)

\[ R=-\left(\sum \theta \omega^2\varphi_i+\theta_{\mathrm I}\omega^2\varphi_{\mathrm I} +\theta_{\mathrm{II}}\omega^2\varphi_{\mathrm{II}}\right)= \]

\[ =-\varphi_1\omega^2(m\theta+\theta_{\mathrm I}+\theta_{\mathrm{II}}) =-\varphi_1\omega^2\sum\theta, \]

or, for \(\varphi_1=1\) radian,

\[ R=-\omega^2\sum\theta, \tag{114} \]

where \(\omega=h\eta\) is the angular velocity of the harmonic moments of the given order acting on the engine shaft;

\(\sum\theta\) is the sum of the moments of inertia of all the flywheel masses of the shafting.

According to formula (19), the harmonic moments \(M\) of the given order cause oscillatory motions of the first mass, and now, under the assumption of absolute rigidity of the shaft and of all the masses of the installation, with amplitude

\[ \varphi_{\mathrm{abs}}=\frac{M}{R}\sum \overrightarrow{\varphi_i} =-\frac{M\sum \overrightarrow{\varphi_i}}{\omega^2\sum\theta}. \tag{115} \]

Here \(\sum \overrightarrow{\varphi_i}\) is the geometric sum of vectors, each of which is equal to 1 radian, found in the phase diagram of the exciting harmonic moments of the given order.

For the principal orders

\[ \sum \overrightarrow{\varphi_i}=m, \]

where \(m\) is the number of cylinders in the engine.

For all other orders, however, in the usual types of multi-cylinder engines having a uniform distribution of flashes in the cylinders with respect to the crankshaft angle of rotation, this sum becomes zero.

Thus, only harmonics of the principal orders create oscillatory motions and determine the nonuniformity of rotation of an absolutely rigid shaft. The minus sign in formula (115)

shows that the forced vibrations of all the masses of the shafting lag by \(180^\circ\) behind the exciting harmonic moments.

The degree of nonuniformity of shaft rotation corresponding to each of these component harmonic oscillatory motions of the entire shafting, regarded as a single rigid body, is now determined, on the basis of formulas (90) and (115), by the expression:

\[ \delta_h = 2\varphi_{\mathrm{abs}}\cdot h = \frac{2Mh\sum \varphi_i}{\omega^2 \sum \theta} = \]

\[ = \frac{2Mhm}{\omega^2\sum \theta} = \frac{2Mm}{h\eta^2\sum \theta}, \tag{116} \]

where \(\eta\), as before, is the angular speed of rotation of the shaft.

Only here, under the assumption of absolute rigidity of the shaft, the degree of nonuniformity \(\delta_h\) is inversely proportional to the square of the angular speed of rotation of the shaft and decreases continuously with an increase in the moment of inertia \(\sum\theta\) of the rotating masses, i.e., with an increase in the moment of inertia of the flywheel and the dynamo rotor.

The values of the classical degree of nonuniformity \(\delta_h\), calculated by formula (116), are shown by the dashed curve in Fig. 18 (in Fig. 18 \(\delta_h\) is denoted by \(\delta_s\)).

The harmonic oscillatory motion of the entire system, caused by harmonics \(M\) of order \(h\), proceeds according to the law:

\[ (\varphi_{\mathrm{abs}})=\varphi_{\mathrm{abs}}\cdot \sin(h\eta t+\beta_h-180^\circ). \]

Here \(\beta_h\) is the phase angle of harmonic \(M\);

\((\varphi_{\mathrm{abs}})\) is the instantaneous angle of deviation (lead or lag) of the shaft relative to uniform rotation with angular speed \(\eta\);

\(\varphi_{\mathrm{abs}}\) is the amplitude of vibration, determined by formula (115).

The corresponding angular velocity of the harmonic oscillatory motion of the shafting is:

\[ (\eta_h) = \frac{d(\varphi_{\mathrm{abs}})}{dt} = \varphi_{\mathrm{abs}}\cdot h\cos(h\eta t+\beta_h-180^\circ) = \]

\[ = \eta_h\cdot \sin(h\eta t+\beta_h-90^\circ), \tag{117} \]

where \(\eta_h=\varphi_{\mathrm{abs}}h\) is the maximum angular velocity of the harmonic motion.

It follows from formula (117) that the velocity vectors \(\eta_h\) lag behind the vectors of the corresponding moments \(M\) by an angle of \(90^\circ\), but, of course, rotate with the same angular velocity \(\eta_h\). The construction

sinusoids (117) for all sufficiently high harmonics \(M\) of the principal orders, and summing their ordinates will give the theoretical tachogram for an absolutely rigid shaft (Fig. 22)\(^{57}\). Laying off downward from the abscissa axis of the constructed tachogram the magnitude of the mean angular velocity \(\eta\), we obtain a new axis \(O_1 t\)—the line of zero rotational speeds of the shaft.

Fig. 22.

Now the classical degree of nonuniformity of rotation of the shaft of the power unit is

\[ \delta_s=\frac{\eta_{\max}-\eta_{\min}}{\eta}, \]

where \(\eta_{\max}\) and \(\eta_{\min}\) are measured from the diagram in Fig. 22.

Of course, this quantity \(\delta_s\) will differ, and often very substantially, from the value obtained when the real elasticity of the shaft and the presence of its torsional vibrations are taken into account.

\(^{57}\) To construct the theoretical tachogram of an absolutely rigid shaft, there is no need to carry out a detailed harmonic analysis of the diagram of tangential forces. It is sufficient, using a harmonic analyzer or, for example, the Fischer-Hinnen or Shprenger method, to determine the amplitudes and phase angles of the harmonics of the principal orders.

VII. ADDITIONAL TORSIONAL STRESSES ARISING IN THE SHAFT DURING ITS TORSIONAL VIBRATIONS

Knowledge of the magnitude of the additional stresses in a shaft arising during its torsional vibrations is urgently necessary for judging the operational reliability of the power installation when it is operating at a prescribed speed condition.

In the practice of design calculations, one sometimes confines oneself only to determining the natural frequencies of the reduced elastic system and, at best, calculates the resonant stresses in the shaft, i.e., those additional torsional stresses that are caused by resonating harmonics of the torque. The calculation of these stresses is not difficult.

The twisting moment in the section of the shaft between masses \(i\) and \(i+1\) is computed by the obvious formula:

\[ M_{i,i+1}=c_{i,i+1}\cdot\left(\varphi_{i+1\,abs}-\varphi_{i\,abs}\right), \]

where \(\varphi_{i\,abs}\) and \(\varphi_{i+1\,abs}\) are the absolute (actual) values of the amplitudes of the forced resonant vibrations of masses \(i\) and \(i+1\), caused by the resonating harmonics;

\(c_{i,i+1}\) is the stiffness of the shaft section between these masses. The quantities \(\varphi_{i\,abs}\) and \(\varphi_{i+1\,abs}\) are determined according to the instructions in Section III.

Now the maximum additional torsional stress in the given section of the shaft is

\[ \tau_{\max}=\frac{M_{i,i+1}}{W_p}, \tag{118} \]

where \(W_p\) is the polar section modulus of the most dangerous cross-section.

To find the most heavily loaded section of the shafting, it is sufficient to construct the curve of the mode shape of the free vibrations of the shafting (see Fig. 9). The segment of this broken curve having the greatest angle of inclination to the abscissa axis corresponds to the shaft section twisted by the greatest additional moment.

But calculations of only the resonant stresses in the shaft do not fully reflect the conditions of its operation, since they do not take into account the influence of other, non-resonating harmonics of the torque and do not at all characterize the loads and stresses in the shaft when the engine is operating outside the critical speeds. Moreover, these calculations do not provide data for determining the magnitude of the removal of the engine operating speeds from its critical speeds—

…of the distance that is necessary to ensure sufficient strength of its shaft when operating at normal speeds.^58

The method described above for constructing theoretical torsiograms and tachograms makes it possible to analyze the additional stresses in the shaft connecting the engine with the flywheel when the engine operates at any number of revolutions, and taking into account any number of orders of the torque harmonics.

The angular velocity of rotation of the flywheel at each instant of time is determined by expression (84), whose differentiation with respect to time gives the angular acceleration of the flywheel

\[ (\eta_{\mathrm{м}}')= \frac{d(\eta_{\mathrm{м}})}{dt} = \sum \eta_h \cdot h \cos(h\eta t+\gamma') = \]

\[ = \sum \eta_h \cdot h \sin(h\eta t+\gamma' + 90^\circ), \]

or, on the basis of formulas (85) and (86),

\[ (\eta_{\mathrm{м}}')= \sum h^2 \eta^2 \varphi_{\mathrm{I}\,\mathrm{abs}} \cdot \sin(h\eta t+\gamma+180^\circ). \tag{119} \]

Hence

\[ (\eta_{\mathrm{м}}')= \sum \eta_{\mathrm{м}}' \cdot \sin(h\eta t+\gamma+180^\circ), \tag{120} \]

where

\[ \eta_{\mathrm{м}}' = h^2\eta^2\varphi_{\mathrm{I}\,\mathrm{abs}} \tag{121} \]

is the maximum angular acceleration of the forced harmonic oscillatory motion of the flywheel caused by harmonic moments of the \(h\)-th order.

Formula (120) shows that the flywheel acceleration vectors, determined by formula (121), lead by \(180^\circ\) the amplitude vectors of its forced oscillations, but have the same angular velocity and frequency.

The summation of the sinusoids

\[ (\eta_{\mathrm{м}}')=\eta_{\mathrm{м}}' \cdot \sin(h\eta t+\gamma+180^\circ), \]

constructed for harmonics of different orders, gives the diagram of the angular accelerations of rotation of the flywheel (Fig. 23).

If all ordinates of this curve are multiplied by the moment of inertia \(\theta_{\mathrm{г}}\) of the flywheel, then the diagram in Fig. 23 will show the variation of the additional twisting moment in the shaft, in the section between the engine and the flywheel, arising as a result of torsional vibrations of the shafting. The maximum

^58 In the work of H. Behrens, published in ZS. für Flugtechnik und Motorluftschiffahrt, 1930, no. 12, p. 297, an approximate method is given for determining the required magnitude of this removal; it is based, however, on a whole series of simplifying assumptions and takes into account the action on the engine shaft of harmonic torques of only one of the principal orders.

Fig. 23

The ordinate of this curve determines the greatest value of the additional twisting moment:

\[ M_{\max}=\theta_I \eta'_{M\max}. \tag{122} \]

Now the greatest additional torsional stress in this section is

\[ \tau_{\max}=\frac{M_{\max}}{W_p}=\frac{\theta_I \eta'_{M\max}}{W_p}. \tag{123} \]

Having constructed the diagrams as in Fig. 23 for various shaft rotation speeds, \(n\), one can construct the curve \(\tau_{\max}=f_2(n)\) and, from it, establish the prohibited (dangerous) and optimal zones of the engine’s operating speeds from the standpoint of ensuring sufficient strength of its shaft between the last cylinder of the engine and the flywheel. This section of the shaft is usually the most dangerous, especially at the last main journal of the crankshaft, since, in addition to the additional moments \(M_{\max}\), it is also loaded by the mean transmitted useful torque \(mM_0\) of the engine, calculated from its power and rotational speed.

Notation

\(A=\dfrac{\Theta\omega^{2}}{c}\) — the principal parameter of the torsional oscillatory process.

\(A_{\mathrm{exc}}\) — the work of the exciting harmonic moments acting on the engine shaft, performed by them over one oscillation cycle.

\(A_{r}\) — the work absorbed by damping forces in the engine, proportional to the oscillation velocity, over one oscillation cycle (calculation according to Wydler).

\(A_{h}\) — the work absorbed by the elastic hysteresis of the shaft over one oscillation cycle (calculation according to Lewis et al.).

\(a_{h}\) — the hysteresis constant in Dorey’s formula.

\(c\) — the stiffness of the crankshaft sections between the axes of each two adjacent engine cylinders, for the system according to Figs. 2 and 3.

\(c_{mI}\) — the stiffness of the shaft between the last engine mass and the first flywheel.

\(c_{I,II}\) — the stiffness of the shaft between the first and second flywheels in the installation according to Fig. 3.

\(c_{i,i+1}\) — the stiffness of the shaft between masses \(i\) and \(i+1\) of the reduced system, in the general case.

\(C_{h}\) — the harmonic coefficient of the \(h\)-th order, in \(\mathrm{kg}/\mathrm{cm}^{2}\), determined by harmonic analysis of the tangential-force diagram for one engine cylinder.

\(d_{a}\) and \(d_{i}\), respectively — the outside and inside diameters of the hollow shaft.

\[ D_{n}=\frac{\varphi_{m}}{\displaystyle\sum_{1}^{m}\varphi_{i}} \]
— an abstract coefficient.

\(F\) — piston area.

\(h\) — the order of the harmonic of the torque on the engine shaft, referred to one revolution of it.

\(i\) — the ordinal number of the installation masses when counting them from the left end of the shaft in Figs. 2 and 3.

\(k\) — the coefficient of velocity resistances in the engine, according to Wydler (in \(\mathrm{kg\,sec}/\mathrm{cm}^{3}\)).

\(k_{0}\) — Shannon damping coefficient (in \(\mathrm{kg\,sec}/\mathrm{cm}\)).

\(l\) — the length of the cylindrical section of the shaft.

\(m\) — the number of engine cylinders.

\(m_{\mathrm{red}}\) — the mass of the mechanism of one cylinder, reduced to the crank radius.

\(M\) — the amplitude of the harmonic moment of the \(h\)-th order, produced on the shaft by one engine cylinder.

\((M)\) — the instantaneous value of this moment.

\(M_{i,i+1}\) — the moment twisting the shaft in the section between masses \(i\) and \(i+1\).

$M_{\max}$ — the maximum moment twisting the shaft in the section between the last engine cylinder and the flywheel.

$M_0$ — the mean torque on the engine shaft, produced by one cylinder.

$N$ — the frequency of oscillation, i.e., the number of oscillation cycles per minute.

$N_0$ — the frequency of the natural oscillations of the installation shaft.

$n$ — the number of revolutions of the engine shaft per minute.

$q$ — the exponent in Dorey’s formula for hysteresis loss in the shaft.

$r$ — the crank radius.

$R$ — the residual moment.

$t$ — time.

$W_p$ — the polar section modulus of the shaft cross section.

$\alpha$ — the angle of rotation of the crank, measured from the top dead center.

$\alpha_M$ — the instantaneous angle of rotation of the flywheel.

$\alpha_I=\dfrac{\theta_I}{\theta}$ — an abstract coefficient.

$\alpha_{II}=\dfrac{\theta_{II}}{\theta}$ — ” ”

$\beta_1=\dfrac{c_{mI}}{c}$ — ” ”

$\beta_2=\dfrac{c_{I,II}}{c}$ — ” ”

$\beta_h$ — the phase angle of the harmonic torque of the $h$-th order.

$\gamma$ — the absolute phase angle of the forced oscillation outside resonance, i.e., the angle between the oscillation vector and the first crank at the initial instant of time, at $t=0$.

$\gamma_0$ — the absolute phase angle of the forced oscillations at resonance.

$\gamma'$ — the absolute phase angle of the velocity vector of the motion of a harmonically oscillating mass.

$\delta$ — the total degree of nonuniformity of rotation of the given mass of the installation, for example, the flywheel.

$\delta_h$ — the degree of nonuniformity of rotation caused by the action on the shaft of harmonic moments of the $h$-th order.

$\delta_S$ — the classical degree of nonuniformity of shaft rotation.

$\varepsilon$ — the relative phase angle of the forced harmonic oscillation of the mass, i.e., the angle between the oscillation vector and the vector of the exciting moment.

$\eta$ — the mean angular velocity of shaft rotation, determined by the formula $\eta=\dfrac{\pi n}{30}\ \mathrm{sec}^{-1}$.

\(\eta_h\) — the maximum angular velocity of the forced harmonic oscillatory motion of the flywheel, produced by harmonic moments of the \(h\)-th order.

\((\eta_h)'\) — the corresponding instantaneous angular velocity of the harmonic motion of the flywheel.

\(\eta_M\) — the actual angular velocity of rotation of the flywheel.

\(\eta'_M\) — the maximum angular acceleration of the flywheel in its harmonic oscillatory motion.

\(\eta'_{M\max}\) — the maximum total angular acceleration of the flywheel.

\(\theta\) — the moment of inertia of the reduced mass of one cylinder of the engine with respect to the shaft axis.

\(\theta_I\) — the moment of inertia of the first flywheel with respect to the shaft axis.

\(\theta_{II}\) — the moment of inertia of the second flywheel (the dynamo rotor) with respect to the same axis.

\(\xi\) — the amplitude of oscillation of the first mass of the engine (in radians), adopted in calculating the hysteresis loss in the shaft according to Rayet or Dorey.

\(\tau_{\max}\) — the maximum torsional stress in the shaft.

\(\tau_c\) — the critical stress according to Dorey.

\(\varphi_1, \varphi_2, \ldots \varphi_i\) — the relative amplitudes of oscillation of the engine masses (with \(\varphi_1 = 1\) radian).

\(\varphi_m\) — the relative amplitude of oscillation of the last mass of the engine.

\(\varphi_I\) and \(\varphi_{II}\) — respectively, the relative amplitudes of oscillation of the first and second flywheels (the dynamo rotor).

\(\varphi_{1\,abs}, \varphi_{2\,abs}, \ldots\)
\(\varphi_{i\,abs}, \varphi_{m\,abs},\)
\(\varphi_{I\,abs}, \varphi_{II\,abs}\) — the absolute amplitudes of the forced harmonic oscillations of the corresponding masses of the system.

\[ \sum_{1}^{m} \varphi_i \]

— the algebraic sum of the relative amplitudes of oscillation of the engine masses.

\[ \sum \varphi_i \]

— the geometric sum of the relative amplitudes of oscillation of the engine masses, determined by constructing the phase diagram of the exciting moments.

\(\omega = h\eta\) — the angular velocity of the forced oscillations produced by harmonic moments of the \(h\)-th order.

\(\omega_0\) — the angular velocity of the natural (free) oscillations of the reduced elastic system.

\[ \Psi = \frac{d_i}{d_a} \]

— the ratio of the inside diameter of a hollow shaft to its outside diameter.

Relative amplitudes of vibrations of the engine masses

\[ \varphi_1=+1{,}0000\ \text{rad.}; \qquad \varphi_2=1-A; \qquad A=\frac{\Theta \omega^2}{c} \]

Table 1

A	φ₃	φ₄	φ₅	φ₆	A	φ₃	φ₄	φ₅	φ₆
0,00	+1,0000	+1,0000	+1,0000	+1,0000	0,27	+0,2629	−0,2752	−0,7390	−1,0032
0,01	+0,9701	+0,9405	+0,9014	+0,8535	0,28	+0,2384	−0,3100	−0,7715	−1,0111
0,02	+0,9404	+0,8820	+0,8059	+0,7138	0,29	+0,2141	−0,3439	−0,8022	−1,0278
0,03	+0,9109	+0,8200	+0,7133	+0,5808	0,30	+0,1900	−0,3770	−0,8309	−1,0355
0,04	+0,8816	+0,7679	+0,6236	+0,4542	0,31	+0,1661	−0,4093	−0,8578	−1,0404
0,05	+0,8525	+0,7124	+0,5366	+0,3341	0,32	+0,1424	−0,4408	−0,8829	−1,0425
0,06	+0,8236	+0,6578	+0,4525	+0,2201	0,33	+0,1189	−0,4714	−0,9062	−1,0419
0,07	+0,7949	+0,6042	+0,3711	+0,1121	0,34	+0,0956	−0,5013	−0,9278	−1,0388
0,08	+0,7664	+0,5515	+0,2925	+0,0100	0,35	+0,0725	−0,5304	−0,9476	−1,0332
0,09	+0,7381	+0,4998	+0,2165	−0,0863	0,36	+0,0496	−0,5587	−0,9658	−1,0253
0,10	+0,7100	+0,4490	+0,1431	−0,1771	0,37	+0,0269	−0,5862	−0,9823	−1,0150
0,11	+0,6821	+0,3992	+0,0723	−0,2625	0,38	+0,0044	−0,6129	−0,9973	−1,0027
0,12	+0,6544	+0,3503	+0,0041	−0,3425	0,39	−0,0179	−0,6388	−1,0106	−0,9882
0,13	+0,6269	+0,3023	−0,0616	−0,4175	0,40	−0,0400	−0,6640	−1,0224	−0,9718
0,14	+0,5996	+0,2553	−0,1248	−0,4874	0,41	−0,0619	−0,6884	−1,0327	−0,9536
0,15	+0,5725	+0,2091	−0,1856	−0,5525	0,42	−0,0836	−0,7120	−1,0415	−0,9335
0,16	+0,5456	+0,1639	−0,2440	−0,6129	0,43	−0,1051	−0,7350	−1,0489	−0,9117
0,17	+0,5189	+0,1196	−0,3001	−0,6687	0,44	−0,1264	−0,7572	−1,0548	−0,8883
0,18	+0,4924	+0,0762	−0,3538	−0,7200	0,45	−0,1475	−0,7786	−1,0594	−0,8634
0,19	+0,4661	+0,0336	−0,4052	−0,7671	0,46	−0,1684	−0,7993	−1,0626	−0,8370
0,20	+0,4400	−0,0080	−0,4544	−0,8099	0,47	−0,1891	−0,8193	−1,0645	−0,8093
0,21	+0,4141	−0,0488	−0,5014	−0,8487	0,48	−0,2096	−0,8386	−1,0651	−0,7803
0,22	+0,3884	−0,0886	−0,5462	−0,8836	0,49	−0,2299	−0,8571	−1,0644	−0,7501
0,23	+0,3629	−0,1277	−0,5889	−0,9146	0,50	−0,2500	−0,8750	−1,0625	−0,7188
0,24	+0,3376	−0,1658	−0,6295	−0,9420	0,51	−0,2699	−0,8922	−1,0594	−0,6864
0,25	+0,3125	−0,2031	−0,6680	−0,9658	0,52	−0,2896	−0,9086	−1,0551	−0,6530
0,26	+0,2876	−0,2396	−0,7045	−0,9862	0,53	−0,3091	−0,9244	−1,0497	−0,6187

Table 1 (continued)

A	φ₃	φ₄	φ₅	φ₆	A	φ₃	φ₄	φ₅	φ₆
0,54	−0,3284	−0,9395	−1,0432	−0,5836	0,86	−0,8404	−1,0981	−0,4114	+0,6291
0,55	−0,3475	−0,9539	−1,0356	−0,5478	0,87	−0,8531	−1,0940	−0,3831	+0,6611
0,56	−0,3664	−0,9676	−1,0270	−0,5112	0,88	−0,8656	−1,0895	−0,3546	+0,6923
0,57	−0,3851	−0,9807	−1,0173	−0,4740	0,89	−0,8779	−1,0845	−0,3259	+0,7228
0,58	−0,4036	−0,9931	−1,0066	−0,4363	0,90	−0,8900	−1,0790	−0,2969	+0,7524
0,59	−0,4219	−1,0049	−0,9950	−0,3980	0,91	−0,9019	−1,0731	−0,2677	+0,7812
0,60	−0,4400	−1,0160	−0,9824	−0,3594	0,92	−0,9136	−1,0667	−0,2384	+0,8092
0,61	−0,4579	−1,0265	−0,9689	−0,3203	0,93	−0,9251	−1,0599	−0,2089	+0,8363
0,62	−0,4756	−1,0363	−0,9545	−0,2809	0,94	−0,9364	−1,0526	−0,1793	+0,8625
0,63	−0,4931	−1,0455	−0,9393	−0,2413	0,95	−0,9475	−1,0449	−0,1496	+0,8878
0,64	−0,5104	−1,0541	−0,9232	−0,2015	0,96	−0,9584	−1,0367	−0,1198	+0,9121
0,65	−0,5275	−1,0621	−0,9064	−0,1615	0,97	−0,9691	−1,0282	−0,0899	+0,9356
0,66	−0,5444	−1,0695	−0,8887	−0,1214	0,98	−0,9796	−1,0192	−0,0560	+0,9580
0,67	−0,5611	−1,0763	−0,8703	−0,0813	0,99	−0,9899	−1,0098	−0,0300	+0,9795
0,68	−0,5776	−1,0824	−0,8512	−0,0412	1,00	−1,0000	−1,0000	−0,0000	+1,0000
0,69	−0,5939	−1,0880	−0,8314	−0,0011	1,01	−1,0099	−0,9898
0,70	−0,6100	−1,0930	−0,8109	+0,0388	1,02	−1,0196	−0,9792
0,71	−0,6259	−1,0974	−0,7898	+0,0786	1,03	−1,0291	−0,9682
0,72	−0,6416	−1,1012	−0,7680	+0,1182	1,04	−1,0384	−0,9569
0,73	−0,6571	−1,1045	−0,7456	+0,1576	1,05	−1,0475	−0,9451
0,74	−0,6724	−1,1072	−0,7227	+0,1966	1,06	−1,0564	−0,9330
0,75	−0,6875	−1,1094	−0,6992	+0,2354	1,07	−1,0651	−0,9205
0,76	−0,7024	−1,1110	−0,6752	+0,2737	1,08	−1,0736	−0,9077
0,77	−0,7171	−1,1120	−0,6507	+0,3117	1,09	−1,0819	−0,8945
0,78	−0,7316	−1,1126	−0,6257	+0,3492	1,10	−1,0900	−0,8810
0,79	−0,7459	−1,1125	−0,6003	+0,3862	1,11	−1,0979	−0,8671
0,80	−0,7600	−1,1120	−0,5744	+0,4227	1,12	−1,1056	−0,8530
0,81	−0,7739	−1,1109	−0,5481	+0,4587	1,13	−1,1131	−0,8384
0,82	−0,7876	−1,1094	−0,5215	+0,4941	1,14	−1,1204	−0,8235
0,83	−0,8011	−1,1073	−0,4944	+0,5288	1,15	−1,1275	−0,8084
0,84	−0,8144	−1,1047	−0,4671	+0,5629	1,16	−1,1344	−0,7929
0,85	−0,8275	−1,1016	−0,4394	+0,5964	1,17	−1,1411	−0,7771

Table 1 (conclusion)

\(A\)	\(\varphi_3\)	\(\varphi_4\)	\(A\)	\(\varphi_3\)	\(\varphi_4\)	\(A\)	\(\varphi_3\)	\(\varphi_4\)
1,18	−1,1476	−0,7610	1,50	−1,2500	−0,1250	1,82	−1,1476	+0,6134
1,19	−1,1539	−0,7447	1,51	−1,2499	−0,1025	1,83	−1,1411	+0,6360
1,20	−1,1600	−0,7280	1,52	−1,2496	−0,0798	1,84	−1,1344	+0,6585
1,21	−1,1659	−0,7111	1,53	−1,2491	−0,0571	1,85	−1,1275	+0,6809
1,22	−1,1716	−0,6931	1,54	−1,2484	−0,0343	1,86	−1,1204	+0,7031
1,23	−1,1771	−0,6764	1,55	−1,2475	−0,0114	1,87	−1,1131	+0,7253
1,24	−1,1824	−0,6586	1,56	−1,2464	+0,0116	1,88	−1,1056	+0,7473
1,25	−1,1875	−0,6406	1,57	−1,2451	+0,0346	1,89	−1,0979	+0,7692
1,26	−1,1924	−0,6224	1,58	−1,2436	+0,0577	1,90	−1,0900	+0,7910
1,27	−1,1971	−0,6039	1,59	−1,2419	+0,0808	1,91	−1,0819	+0,8126
1,28	−1,2016	−0,5852	1,60	−1,2400	+0,1040	1,92	−1,0736	+0,8341
1,29	−1,2059	−0,5662	1,61	−1,2379	+0,1272	1,93	−1,0651	+0,8554
1,30	−1,2100	−0,5470	1,62	−1,2356	+0,1505	1,94	−1,0564	+0,8766
1,31	−1,2139	−0,5276	1,63	−1,2331	+0,1738	1,95	−1,0475	+0,8976
1,32	−1,2176	−0,5080	1,64	−1,2304	+0,1971	1,96	−1,0384	+0,9185
1,33	−1,2211	−0,4881	1,65	−1,2275	+0,2204	1,97	−1,0291	+0,9391
1,34	−1,2244	−0,4681	1,66	−1,2244	+0,2437	1,98	−1,0196	+0,9596
1,35	−1,2275	−0,4479	1,67	−1,2211	+0,2670	1,99	−1,0099	+0,9799
1,36	−1,2304	−0,4275	1,68	−1,2176	+0,2904	2,00	−1,0000	+1,0000
1,37	−1,2331	−0,4069	1,69	−1,2139	+0,3137
1,38	−1,2356	−0,3861	1,70	−1,2100	+0,3370
1,39	−1,2379	−0,3651	1,71	−1,2059	+0,3603
1,40	−1,2340	−0,3410	1,72	−1,2016	+0,3836
1,41	−1,2419	−0,3227	1,73	−1,1971	+0,4068
1,42	−1,2436	−0,3013	1,74	−1,1924	+0,4300
1,43	−1,2451	−0,2797	1,75	−1,1875	+0,4531
1,44	−1,2464	−0,2580	1,76	−1,1824	+0,4762
1,45	−1,2475	−0,2361	1,77	−1,1771	+0,4993
1,46	−1,2484	−0,2141	1,78	−1,1716	+0,5222
1,47	−1,2491	−0,1920	1,79	−1,1659	+0,5452
1,48	−1,2496	−0,1698	1,80	−1,1600	+0,5680
1,49	−1,2498	−0,1474	1,81	−1,1539	+0,5908

$\varphi_1=+1.0000$ rad. Values of $\displaystyle\sum_1^4 \varphi_i$ and $D_4$ for a four-cylinder engine

Table II

$\Delta$	$\displaystyle\sum_1^4 \varphi_i$	$D_4$	$\Delta$	$\displaystyle\sum_1^4 \varphi_i$	$D_4$	$\Delta$	$\displaystyle\sum_1^4 \varphi_i$	$D_4$	$\Delta$	$\displaystyle\sum_1^4 \varphi_i$	$D_4$
0.00	+4.0000	+0.2500	0.27	+1.7177	−0.1602	0.54	+0.1921	−4.8893	0.81	−0.6948	+1.5988
0.01	+3.9006	+0.2411	0.28	+1.6484	−0.1880	0.55	+0.1486	−6.4180	0.82	−0.7170	+1.5473
0.02	+3.8024	+0.2320	0.29	+1.5802	−0.2176	0.56	+0.1060	−9.1298	0.83	−0.7384	+1.4996
0.03	+3.7009	+0.2216	0.30	+1.5130	−0.2492	0.57	+0.0642	−15.2739	0.84	−0.7591	+1.4553
0.04	+3.6095	+0.2128	0.31	+1.4468	−0.2829	0.58	+0.0233	−42.6448	0.85	−0.7791	+1.4139
0.05	+3.5149	+0.2027	0.32	+1.3816	−0.3191	0.59	−0.0168	+59.8891	0.86	−0.7985	+1.3752
0.06	+3.4214	+0.1923	0.33	+1.3175	−0.3578	0.60	−0.0560	+18.1429	0.87	−0.8171	+1.3389
0.07	+3.3291	+0.1815	0.34	+1.2543	−0.3997	0.61	−0.0944	+10.8759	0.88	−0.8351	+1.3046
0.08	+3.2379	+0.1709	0.35	+1.1921	−0.4449	0.62	−0.1319	+7.8533	0.89	−0.8524	+1.2723
0.09	+3.1479	+0.1588	0.36	+1.1309	−0.4940	0.63	−0.1686	+6.1996	0.90	−0.8690	+1.2412
0.10	+3.0590	+0.1468	0.37	+1.0707	−0.5474	0.64	−0.2045	+5.1536	0.91	−0.8850	+1.2125
0.11	+2.9713	+0.1344	0.38	+1.0115	−0.6059	0.65	−0.2396	+4.4324	0.92	−0.9003	+1.1848
0.12	+2.8847	+0.1214	0.39	+0.9533	−0.6701	0.66	−0.2739	+3.9048	0.93	−0.9150	+1.1584
0.13	+2.7992	+0.1080	0.40	+0.8960	−0.7411	0.67	−0.3073	+3.5016	0.94	−0.9290	+1.1330
0.14	+2.7149	+0.0940	0.41	+0.8397	−0.8199	0.68	−0.3400	+3.1833	0.95	−0.9424	+1.1088
0.15	+2.6316	+0.0795	0.42	+0.7843	−0.9079	0.69	−0.3719	+2.9255	0.96	−0.9551	+1.0854
0.16	+2.5495	+0.0643	0.43	+0.7299	−1.0070	0.70	−0.4030	+2.7122	0.97	−0.9673	+1.0630
0.17	+2.4685	+0.0484	0.44	+0.6764	−1.1194	0.71	−0.4333	+2.5326	0.98	−0.9788	+1.0413
0.18	+2.3886	+0.0319	0.45	+0.6239	−1.2480	0.72	−0.4628	+2.3793	0.99	−0.9897	+1.0203
0.19	+2.3097	+0.0146	0.46	+0.5723	−1.3968	0.73	−0.4916	+2.2367	1.00	−1.0000	+1.0000
0.20	+2.2320	−0.0036	0.47	+0.5216	−1.5709	0.74	−0.5196	+2.1308	1.01	−1.0097	+0.9803
0.21	+2.1553	−0.0226	0.48	+0.4718	−1.7724	0.75	−0.5469	+2.0286	1.02	−1.0188	+0.9611
0.22	+2.0798	−0.0426	0.49	+0.4230	−2.0266	0.76	−0.5734	+1.9576	1.03	−1.0273	+0.9425
0.23	+2.0052	−0.0637	0.50	+0.3750	−2.3333	0.77	−0.5991	+1.8561	1.04	−1.0353	+0.9262
0.24	+1.9318	−0.0858	0.51	+0.3279	−2.7205	0.78	−0.6242	+1.7825	1.05	−1.0426	+0.9065
0.25	+1.8597	−0.1092	0.52	+0.2818	−3.2244	0.79	−0.6484	+1.7157	1.06	−1.0494	+0.8891
0.26	+1.7880	−0.1340	0.53	+0.2365	−3.9843	0.80	−0.6720	+1.6548	1.07	−1.0556	+0.8720

Table II (conclusion)

\(A\)	\(\sum_1^4 \varphi_i\)	\(D_4\)	\(A\)	\(\sum_1^4 \varphi_i\)	\(D_4\)	\(A\)	\(\sum_1^4 \varphi_i\)	\(D_4\)
1,08	−1,0613	+0,8553	1,39	−0,9930	+0,3677	1,70	−0,5730	−0,5881
1,09	−1,0664	+0,8388	1,40	−0,9780	+0,3517	1,71	−0,5556	−0,6485
1,10	−1,0710	+0,8226	1,41	−0,9746	+0,3311	1,72	−0,5330	−0,7129
1,11	−1,0750	+0,8066	1,42	−0,9649	+0,3123	1,73	−0,5203	−0,7818
1,12	−1,0785	+0,7908	1,43	−0,9548	+0,2929	1,74	−0,5024	−0,8558
1,13	−1,0815	+0,7752	1,44	−0,9444	+0,2732	1,75	−0,4841	−0,9355
1,14	−1,0839	+0,7598	1,45	−0,9336	+0,2529	1,76	−0,4662	−1,0216
1,15	−1,0859	+0,7444	1,46	−0,9225	+0,2321	1,77	−0,4478	−1,1149
1,16	−1,0873	+0,7292	1,47	−0,9111	+0,2108	1,78	−0,4294	−1,2164
1,17	−1,0882	+0,7141	1,48	−0,8994	+0,1888	1,79	−0,4118	−1,3240
1,18	−1,0886	+0,6991	1,49	−0,8873	+0,1662	1,80	−0,3920	−1,4490
1,19	−1,0886	+0,6841	1,50	−0,8750	+0,1429	1,81	−0,3731	−1,5832
1,20	−1,0880	+0,6691	1,51	−0,8624	+0,1188	1,82	−0,3542	−1,7320
1,21	−1,0870	+0,6542	1,52	−0,8494	+0,0940	1,83	−0,3351	−1,8981
1,22	−1,0847	+0,6390	1,53	−0,8362	+0,0683	1,84	−0,3159	−2,0845
1,23	−1,0835	+0,6243	1,54	−0,8227	+0,0416	1,85	−0,2966	−2,2954
1,24	−1,0810	+0,6093	1,55	−0,8089	+0,0141	1,86	−0,2773	−2,5361
1,25	−1,0778	+0,5944	1,56	−0,7948	−0,0146	1,87	−0,2578	−2,8134
1,26	−1,0748	+0,5791	1,57	−0,7805	−0,0443	1,88	−0,2383	−3,1364
1,27	−1,0710	+0,5639	1,58	−0,7659	−0,0753	1,89	−0,2187	−3,5178
1,28	−1,0668	+0,5485	1,59	−0,7511	−0,1076	1,90	−0,1990	−3,9749
1,29	−1,0621	+0,5331	1,60	−0,7360	−0,1413	1,91	−0,1793	−4,5330
1,30	−1,0570	+0,5175	1,61	−0,7207	−0,1765	1,92	−0,1595	−5,2299
1,31	−1,0515	+0,5018	1,62	−0,7051	−0,2134	1,93	−0,1397	−6,1253
1,32	−1,0455	+0,4868	1,63	−0,6893	−0,2521	1,94	−0,1198	−7,3183
1,33	−1,0392	+0,4697	1,64	−0,6733	−0,2927	1,95	−0,0998	−8,9875
1,34	−1,0325	+0,4534	1,65	−0,6571	−0,3354	1,96	−0,0798	−11,4900
1,35	−1,0254	+0,4368	1,66	−0,6407	−0,3804	1,97	−0,0600	−15,6591
1,36	−1,0179	+0,4200	1,67	−0,6241	−0,4279	1,98	−0,0400	−25,9950
1,37	−1,0100	+0,4028	1,68	−0,6072	−0,4782	1,99	−0,0200	−48,9975
1,38	−1,0017	+0,3854	1,69	−0,5902	−0,5315	2,00	−0,0000	−∞

\[ \varphi_1=+1{,}0000\ \text{rad.} \]

Values of \(\displaystyle \sum_{1}^{6}\varphi_i\) and \(D_6\) for a six-cylinder engine

Table II

\(A\)	\(\displaystyle \sum_{1}^{6}\varphi_i\)	\(D_6\)	\(A\)	\(\displaystyle \sum_{1}^{6}\varphi_i\)	\(D_6\)	\(A\)	\(\displaystyle \sum_{1}^{6}\varphi_i\)	\(D_6\)	\(A\)	\(\displaystyle \sum_{1}^{6}\varphi_i\)	\(D_6\)
0,00	+6,0000	+0,1667	0,27	−0,0245	+40,9833	0,54	−1,4347	+0,4068	0,81	−0,7843	−0,5848
0,01	+5,6555	+0,1509	0,28	−0,1401	+7,2581	0,55	−1,4548	+0,3818	0,82	−0,7444	−0,6637
0,02	+5,3221	+0,1341	0,29	−0,2497	+4,1156	0,56	−1,4322	+0,3569	0,83	−0,7040	−0,7511
0,03	+4,9949	+0,1163	0,30	−0,3534	+2,9209	0,57	−1,4271	+0,3322	0,84	−0,6632	−0,8487
0,04	+4,6873	+0,0965	0,31	−0,4514	+2,3049	0,58	−1,4196	+0,3073	0,85	−0,6221	−0,9585
0,05	+4,3856	+0,0762	0,32	−0,5437	+1,9172	0,59	−1,4098	+0,2823	0,86	−0,5808	−1,0832
0,06	+4,0940	+0,0538	0,33	−0,6307	+1,6521	0,60	−1,3978	+0,2571	0,87	−0,5392	−1,2261
0,07	+3,8123	+0,0294	0,34	−0,7123	+1,4584	0,61	−1,3836	+0,2315	0,88	−0,4974	−1,3919
0,08	+3,5404	+0,0028	0,35	−0,7887	+1,3100	0,62	−1,3674	+0,2054	0,89	−0,4555	−1,5869
0,09	+3,2780	−0,0263	0,36	−0,8602	+1,1920	0,63	−1,3492	+0,1788	0,90	−0,4135	−1,8197
0,10	+3,0250	−0,0585	0,37	−0,9266	+1,0954	0,64	−1,3292	+0,1516	0,91	−0,3715	−2,1029
0,11	+2,7811	−0,0944	0,38	−0,9884	+1,0144	0,65	−1,3075	+0,1235	0,92	−0,3295	−2,4557
0,12	+2,5462	−0,1345	0,39	−1,0456	+0,9452	0,66	−1,2840	+0,0945	0,93	−0,2876	−2,9076
0,13	+2,3201	−0,1799	0,40	−1,0982	+0,8849	0,67	−1,2590	+0,0646	0,94	−0,2458	−3,5083
0,14	+2,1026	−0,2318	0,41	−1,1466	+0,8317	0,68	−1,2324	+0,0334	0,95	−0,2042	−4,3472
0,15	+1,8935	−0,2918	0,42	−1,1907	+0,7840	0,69	−1,2044	+0,0009	0,96	−0,1628	−5,6027
0,16	+1,6926	−0,3621	0,43	−1,2307	+0,7408	0,70	−1,1751	−0,0330	0,97	−0,1216	−7,6916
0,17	+1,4997	−0,4459	0,44	−1,2667	+0,7013	0,71	−1,1445	−0,0687	0,98	−0,0808	−11,8638
0,18	+1,3148	−0,5477	0,45	−1,2989	+0,6647	0,72	−1,1126	−0,1062	0,99	−0,0402	−24,3694
0,19	+1,1375	−0,6744	0,46	−1,3273	+0,6306	0,73	−1,0797	−0,1459	1,00	−0,0000	−∞
0,20	+0,9681	−0,8366	0,47	−1,3522	+0,5985	0,74	−1,0457	−0,1880
0,21	+0,8052	−1,0540	0,48	−1,3736	+0,5681	0,75	−1,0107	−0,2329
0,22	+0,6500	−1,3594	0,49	−1,3915	+0,5390	0,76	−0,9749	−0,2808
0,23	+0,5017	−1,8230	0,50	−1,4062	+0,5111	0,77	−0,9382	−0,3322
0,24	+0,3603	−2,6144	0,51	−1,4178	+0,4841	0,78	−0,9007	−0,3877
0,25	+0,2256	−4,2814	0,52	−1,4263	+0,4578	0,79	−0,8625	−0,4478
0,26	+0,0974	−10,1280	0,53	−1,4319	+0,4321	0,80	−0,8237	−0,5132

The dirty, ragged printing of the Greek letter “sigma” occurred through no fault of the printing house. Owing to the lack of a factory-made “sigma,” the latter was produced by a handicraft method.

Contents

		Page
I.	Introduction	3
II.	Forced vibrations outside resonance	13
III.	Forced vibrations at resonance	23
IV.	Theoretical torsiograms and the actual degree of nonuniformity of rotation	51
V.	Critical and optimal values of \(CD^2\) of the flywheel	60
VI.	Classical degree of nonuniformity of rotation	68
VII.	Additional torsional stresses in the shaft arising during its torsional vibrations	72

Responsible editor: Prof. M. A. Usov.
Technical editor: N. Dugina.

Submitted for production 4/VII 1937.
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1. The possibility of such cases was pointed out earlier by H. Lorenz, Dynamik der Kurbelgetriebe, Leipzig, 1901, and A. Sommerfeld, “Z. d. V. D. I.,” 1904, p. 631; see also the work by K. Röhrich cited in footnote 5. ↩

2. F. Vogt, “Über schädliche Schwungmassen bei Drehschwingungen,” “Z. d. V. D. I.,” 1927, p. 1221. The concept of the critical value of the flywheel moment was introduced for the first time by H. Görges and E. Rosenberg in works on the theory of vibrations of parallel-connected alternating-current machines. See also F. Punga, “E. T. Z.,” p. 385, 1911. But in these works torsional vibrations of shafts were not taken into account at all. ↩

3. H. Wydler, Drehschwingungen in Kolbenkraftanlagen, Berlin, 1922.
   V. Zhitomirskii, “The Influence of Torsional Vibrations of the Shaft on the Degree of Nonuniformity of a Diesel Engine,” Proceedings of the All-Union Heat Engineering Institute, 1930, Nos. 6–7 (59).
   V. Zhitomirskii, “The True Degree of Nonuniformity,” Diesel Engineering, 1934, No. 1. ↩

4. V. Nechaev, “Shift of Critical Speeds,” Proceedings of the Tomsk Industrial Institute, vol. LVI, 1936.
   V. Gudkov-Belyakov, “Displacement of Critical Numbers of Revolutions in Piston Engines,” Diesel Engineering, 1935, No. 7. ↩

5. F. Porter, “Trans. A.S.M.E.,” Vol. 50, 1927—1928. ↩

6. In a multi-mass system having several natural frequencies, it is in principle possible for several harmonic components of the torque to enter into resonance simultaneously. ↩

7. H. Holzer, Die Berechnung der Drehschwingungen, Berlin, 1922. F. Porter, “Trans. A. S. M. E.,” Vol. L. ↩

8. Rayleigh, Theorie des Schalles, Braunschweig, 1880, Bd. 1. ↩

9. This relation may be obtained on the basis of a general analysis of torsional vibrations without applying Rayleigh’s theorem; see the above-cited work of Holzer, p. 130, and also M. Tolle, Regelung der Kraftmaschinen, 3. Aufl., Berlin, 1921. ↩

10. Theoretical torsiograms usually have to be constructed for the flywheel or the rotor of the dynamo, i.e., for the extreme right-hand, according to Figs. 2 and 3, mass of the reduced elastic system. This considerably facilitates the solution of our problem, since it eliminates the necessity of a detailed calculation of the forced nonresonant vibrations of all the masses of the installation. ↩

11. V. K. Nechaev, “A graphical method for determining natural frequencies,” Proceedings of the Tomsk Industrial Institute, vol. LIV, issue VI, 1935, and also Diesel Engineering, 1934, No. 4. ↩

12. Tables II and III also give the values \(D_4\) and \(D_6\), required in calculating the natural frequencies of installations with four- and six-cylinder engines by the method set forth by the author in his above-cited article “A Graphical Method for Determining Natural Frequencies.” ↩

13. For the system of Fig. 3, instead of \(\Phi_{1,\mathrm I abs}\) one should write \(\Phi_{1,\mathrm{II} abs}\), etc. In what follows, for brevity of exposition, we shall consider the installation of Fig. 2. ↩

14. Such a refinement is necessary, for example, in investigating the effect of cutting out individual cylinders of an engine on the torsional vibrations of its shaft. ↩

15. In this case the angle \(\varepsilon=90^\circ\) is preserved only between the harmonic moment \(M_i\) and the oscillation it causes at the point of its application. ↩

16. See his work cited in footnote 25. ↩

17. The opposition of phases of vibration of individual masses is taken into account by the sign of \(\varphi_i\). ↩

18. That is, resistance forces proportional to the velocity of the oscillatory motion of the masses. ↩

19. On the method of determining damping forces on a propeller, see F. Lewis, “Trans. Soc. Nav. Arch. and Mar. Eng.,” 1925; F. Porter, “Trans. A. S. M. E.,” Vol. 50. ↩

20. H. Wydler, op. cit., p. 55. ↩

21. F. Rowett, Elastic hysteresis in steel. “Proc. of Roy. Soc. of London”, Vol. 88, p. 528. ↩

22. O. FöppI, Grundzüge der technischen Schwingungslehre, Berlin 1931.
    F. Porter, “Trans. A. S. M. E.,” Vol. 50. ↩

23. J. P. den Hartog and J. Ormondroyd, “Torsional-vibration dampers,” Trans. A.S.M.E., Vol. 52 (1930). ↩

24. L. Geiger, Mechanische Schwingungen, 1927.
    V. Zhitomirskii, “Izv. V. T. I.,” 1930, No. 6–7. ↩

25. Here \((\eta_h)\), in contrast to \(\eta_h\), denotes the instantaneous, and not the maximum, value of the angular velocity of the harmonic oscillatory motion of the mass. ↩